Abstract
From the standard model of particle physics to strongly correlated electrons, various physical settings are formulated in terms of matter coupled to gauge fields. Quantum simulations based on ultracold atoms in optical lattices provide a promising avenue to study these complex systems and unravel the underlying many-body physics. Here, we demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices, using a combination of coherent lattice modulation with strong interactions. Specifically, we propose implementation of ℤ2 lattice gauge theories coupled to matter, reminiscent of theories previously introduced in high-temperature superconductivity. We discuss a range of settings from zero-dimensional toy models to ladders featuring transitions in the gauge sector to extended two-dimensional systems. Mastering lattice gauge theories in optical lattices constitutes a new route toward the realization of strongly correlated systems, with properties dictated by an interplay of dynamical matter and gauge fields.
INTRODUCTION
Gauge fields play a central role in a wide range of physical settings: The interactions in the standard model are mediated by gauge bosons, and everyday phenomena related to electromagnetism are governed by Maxwell’s equations featuring a gauge symmetry. The presence of strong magnetic fields can lead to strong alterations of the behavior of interacting many-body systems; for example, in the fractional quantum Hall (FQH) effect, the statistics of elementary excitations can be transmuted from fermionic to bosonic or to neither of both (anyonic) (1). Last, gauge theories even play a role in strongly correlated quantum systems, where local constraints lead to emergent gauge symmetries at low energies; for example, frustrated quantum spin liquids can be classified by their corresponding gauge theories.
The realization of artificial gauge fields in ultracold gases is an important milestone, enabling studies of the interplay between gauge fields and strong interactions in quantum many-body systems. This feat has further promoted these quantum-engineered systems as versatile quantum simulators (2, 3). While a synthetic magnetic field can be simply introduced by rotating atomic clouds (4, 5), more sophisticated schemes were developed to generate a wide family of gauge field structures, including spin-orbit couplings (6) or patterns featuring staggered magnetic fluxes with alternating signs on a length scale given by the lattice constant (7–9). The design of magnetic fluxes for ultracold atoms in optical lattices, through laser-induced tunneling or shaking methods, has been recently exploited in view of realizing topological states of matter (10, 11) and frustrated magnetism (8). In the settings described above, artificial gauge fields are treated as classical and nondynamical; however, in the sense that they remain insensitive to the spatial configuration and motion of the atomic cloud, these engineered systems do not aim to reproduce a complete gauge theory, where particles and gauge fields influence each other.
To be able to use ultracold atoms and simulate a wider range of physical problems, including those from high-energy physics, two major steps need to be taken. First, the synthetic gauge fields need to be made intrinsically dynamical, allowing back-actions of the particles on the gauge field. For example, the strength of the synthetic magnetic field may depend explicitly on the local particle density. In a second step, the dynamics of the synthetic gauge fields needs to be constrained to fulfill certain local symmetries. Therefore, the synthetic gauge field interacts with the matter particles, but each lattice site is also associated with a separately conserved charge. Theories of this type are called lattice gauge theories (LGTs), and the detailed conservation laws they satisfy depend on the respective gauge group (12). The simplest instant of an LGT has a ℤ2, or Ising, gauge group, but in the presence of fermionic matter, even this model poses a substantial theoretical challenge (13).
Various theoretical works have already suggested several methods by which synthetic gauge fields can be made intrinsically dynamical. A first approach builds on the rich interplay between laser-induced tunneling and strong on-site interactions, which can be both present and finely controlled in an optical lattice (3): Under specific conditions, the tunneling matrix elements, which not only describe the hopping on the lattice but also capture the presence of a gauge field, can become density dependent (14–18); see (19) for an experimental implementation of these density-dependent gauge fields. While these settings include rich physics, they lack local conservation laws and thus still differ significantly from problems relevant to, e.g., high-energy physics.
A second approach aims at directly implementing genuine LGTs with local conservation laws, such as the Kogut-Susskind or quantum link models. This can be achieved, in principle, by engineering specific model Hamiltonians through elaborate laser-coupling schemes involving different atomic species and well-designed constraints; see (20–22) for reviews and (23) for an ion trap realization of the Kogut-Susskind Hamiltonian. A digital implementation of ℤ2 LGTs, including couplings to fermionic matter, was proposed in (24). These quantum simulations of LGTs aim to deepen our understanding of fundamental concepts of gauge theories (20–23, 25), such as confinement and its interplay with dynamical charges, which are central in high-energy (26) and condensed-matter physics (13, 27, 28) and go beyond a mere density dependence of synthetic gauge fields. While important first steps have been taken, the direct quantum simulation of LGTs is still in its infancy. The implementations proposed so far still contain significant technical challenges that need to be overcome, making alternative implementation schemes desirable.
In this work, we demonstrate how ℤ2 LGTs can be realized in ultracold gases through the use of specifically designed density-dependent gauge fields. Our approach combines the experimental advantages afforded by settings with density-dependent synthetic gauge fields and the additional physical structure added by the presence of local conservation laws. We demonstrate how existing ultracold atom technology can be used to implement toy models relevant to both high-energy and condensed-matter physics, and describe how the procedure can be scaled up to transition from simplistic two-site models to two-dimensional (2D) systems with direct relevance to studies of, e.g., high-temperature superconductivity (13).
As a central ingredient, we devise a scheme to engineer flux attachment for cold atoms moving in an optical lattice. Originally introduced by Wilczek (29, 30), and then widely exploited in the context of the FQH effect (1), flux attachment is a mathematical construction according to which a certain amount of magnetic flux quanta is attached to a particle (e.g., an electron). The resulting composite “flux tube particle” can change its quantum statistics from bosonic to fermionic, or vice versa (30), and naturally appears in field theoretical formulations of FQH states (1). Specifically, we show that an optical lattice loaded with two atomic species (a and f) can be configured in a way that a localized f-particle becomes a source of magnetic flux Φ for the a-particle: The magnetic flux can thus effectively be attached to the f-particles, which are also allowed to move around the lattice (see Fig. 1A). (Cases with only one species, more closely related to the FQH effect, can also be considered.) The flux attachment scheme is our starting point for implementing ℤ2 LGTs using ultracold atoms.
(A) We propose a setup where one atomic species f becomes a source of magnetic flux Φ (red) for a second species a. Both types of atoms undergo coherent quantum dynamics, described by NN tunneling matrix elements ta and tf, respectively. (B) When realized in a ladder geometry, the flux attachment setup has a ℤ2 lattice gauge structure. By tuning the ratio of the tunneling elements ta/tf, we find that the system undergoes a phase transition. The two regimes can be understood in terms of the elementary ingredients of a ℤ2 LGT, summarized in (C). The matter field
For specific choices of parameters and carefully designed lattice geometries, we first show that this appealing setting can be readily used to implement interacting quantum systems with ℤ2 link variables and global ℤ2 gauge symmetries. Then, we demonstrate that our method can also be extended to systems with local symmetries, realizing genuine ℤ2 LGTs (26) in various lattice geometries. These latter types of models, where the matter field couples to a ℤ2 lattice gauge field, are especially relevant in the context of high-Tc superconductivity (13, 31) and, more generally, strongly correlated electrons (32–34). A central question in this context concerns the possibility of a confinement-deconfinement transition in the LGT (12), which would indicate electron fractionalization (13, 35, 36). The proposed model will allow us to explore the interplay of a global U(1) symmetry with local ℤ2 symmetries, which has attracted particular attention in the context of high-Tc cuprate compounds (34, 37, 38).
We discuss in detail the physics of a toy model characterized by a global U(1) × ℤ2 symmetry, which consists of a two-leg ladder geometry and can be directly accessed with state-of-the-art cold atom experiments. We demonstrate that the toy model features an intricate interplay of matter and gauge fields, as a result of which the system undergoes a phase transition in the ℤ2 sector depending on the ratio of the species-dependent tunnel couplings ta/tf (see Fig. 1B). While this transition can be characterized by the spontaneously broken global ℤ2 symmetry, we argue that an interpretation in terms of the constituents of a ℤ2 LGT (see Fig. 1C) is nevertheless useful to understand its microscopic origin. We also predict a phase transition of the matter field from an insulating Mott state to a gapless superfluid (SF) regime, associated with the spontaneously broken global U(1) symmetry. For appropriate model parameters, an interplay of both types of transitions can be observed.
The paper is organized as follows. We start by introducing the flux attachment scheme, which is at the heart of the proposed experimental implementation of dynamical gauge fields. Particular attention is devoted to the case of a double-well system, which forms the common building block for realizing ℤ2 LGTs coupled to matter. Next, we study the phase diagram of a toy model with a two-leg ladder geometry, consisting of a matter field coupled to a ℤ2 gauge field on the rungs. Realistic implementations of the considered models are proposed afterward, along with a scheme for realizing genuine ℤ2 LGTs with local instead of global symmetries in two dimensions. This paves the way for future investigations of strongly correlated systems, as discussed in the summary and outlook section.
The minimal model of a ℤ2 LGT coupled to matter proposed here has been realized experimentally in a double-well system (39). Besides, density-dependent Peierls phases have been realized with two-component fermions in (40), based on a two-frequency driving scheme, which is proposed below as an ingredient to implement ℤ2 LGTs in extended lattices.
RESULTS
Flux attachment
The recent experimental implementations of classical gauge fields for ultracold atoms (41–45) combine two key ingredients (46): First, the bare tunnel couplings t are suppressed by large energy offsets ∣Δ∣≫ t, realized by a magnetic field gradient or a superlattice potential. Second, tunneling is restored with complex phases ϕ by proper time modulation of the optical lattice (47, 48) at the resonance frequency ω = Δ (with ħ = 1 throughout). The phase of the lattice shaking directly determines the value ϕ of the complex hopping element.
Flux attachment operates in a strongly correlated regime, where the energy offsets Δ = ω from an external potential are supplemented by interspecies Hubbard interactions of the same magnitude, U = ω (49). This provides coherent control over the synthetic gauge fields induced by the lattice modulation at frequency ω [see also (14–18)].
We consider a situation where atoms of a first species, with annihilation operators
Model. The largest energy scale in our problem is set by strong interspecies Hubbard interactions
A minimal example is illustrated in Fig. 2A.
(A) We consider a double-well setup with one atom of each type, a and f. Coherent tunneling between the two orbitals at j1 and j2 = j1 + ey is suppressed for both species by strong Hubbard interactions U = ω, and for f-particles by the energy offset Δf = ω indicated by the blue triangle. (B) Tunnel couplings can be restored by resonant lattice modulations with frequency ω. The sign of the restored tunneling matrix element is different when the a-particle gains (top) or loses (bottom) energy. (C) This difference in sign gives rise to a ℤ2 gauge structure and allows the implementation of ℤ2 minimal coupling of the matter field
Coherent dynamics of both fields are introduced by NN tunneling matrix elements in the μ = x, y directions,
To restore tunnel couplings with complex phases, we include a time-dependent lattice modulation
It acts equally on both species and is periodic in time, Vω(j, t + 2π/ω) = Vω(j, t), with frequency ω = U resonant with the interspecies interactions. In summary, our Hamiltonian is
Effective hopping Hamiltonian. From now on, we consider resonant driving,
The Hermitian operators
Explicit expressions for
Here 𝒥n denotes the Bessel function of the first kind, x = Ai,j/ω is the dimensionless driving strength, and
Without loss of generality, we assume ω, Ai,j > 0 throughout the paper.
The complex phases of the restored tunnelings are also determined by the many-body energy offsets Δrs = nrsω. If nrs ≥ 0, the particle gains energy in the hopping process and
In contrast, if nrs < 0, the particle loses energy and
In this case, there is an additional nrsπ phase shift due to the reflection properties of the Bessel function, Jn(−x) = (−1)nJn(x) (see Fig. 2B). This nrs π phase shift is at the core of the LGT implementations discussed below. Similar results are obtained for
As illustrated in Fig. 1A, our scheme allows us to implement effective Hamiltonians (Eq. 7) describing a mixture of two species, where one acts as a source of magnetic flux for the other [see also (17)]. A detailed discussion of the resulting Harper-Hofstadter model with dynamical gauge flux is provided in section S1. By analogy with the physics of the FQH effect (50, 51), we expect that this flux attachment gives rise to interesting correlations and possibly to quasiparticle excitations with nontrivial statistics.
ℤ2 LGT in a double well. Now, we apply the result in Eq. 7 and discuss a minimal setting, where one a-particle and one f-particle each tunnel between the two sites j1 and j2 = j1 + ey of a double-well potential (see Fig. 2A); ey denotes the unit vector along y. This system forms the central building block for the implementation of ℤ2 LGTs in larger systems, proposed below. We assume that Va(ji) ≡ 0 for i = 1,2 but introduce a potential offset Vf(j2) = Δf + Vf(j1) for the f species, breaking the symmetry between a- and f-particles.
Effective Hamiltonian
For Δf = U = ω and lattice modulations with a trivial phase ϕj1,j2 = 0, the effective Floquet Hamiltonian in Eq. 7 becomes
As shown in Fig. 2B, the interaction energy of the matter field changes by ±U in every tunneling event. As a result, the amplitude renormalization in Eq. 12 is λy = ∣ 𝒥1(Aj2,j1/ω)∣ (see Eq. 8), and the phase of the restored tunnel couplings is
The effective Hamiltonian (Eq. 12) realizes a minimal version of a ℤ2 LGT: The link variable
Symmetries
Each of the two lattice sites ji is associated with a ℤ2 symmetry. The operators generating the ℤ2 gauge group in the double-well system
Physically, Eq. 15 establishes a relation between the ℤ2 electric field lines,
The model in Eq. 12 is invariant under the gauge symmetries
In this case, neither of the amplitude renormalizations
Intuition
We take this opportunity to explain in a mechanistic way the meaning of the ℤ2 gauge field in the double-well system and its relation to more general LGTs. As a starting point, consider the situation when
The two possible states of the link variable, corresponding to the two positions j1 and j2 of the f atom, define a 2D Hilbert space on the link 〈j2, j1〉. This Hilbert space is equivalent to the Hilbert space of a ℤ2 lattice gauge field, with two orthogonal states on each link of the lattice. Using this language, we can identify the operator
If we allow to add arbitrary perturbations to the tunneling Hamiltonian, e.g., terms
A genuine ℤ2 LGT is obtained if only terms are included in the Hamiltonian, which commute with the ℤ2 gauge operators
Matter gauge field coupling in two-leg ladders
In the following, we study the physics of coupled matter and gauge fields in a two-leg ladder, accessible with numerical density matrix renormalization group (DMRG) simulations (54). Our starting point is a model with minimal couplings to the ℤ2 gauge field on the rungs of the ladder, which is characterized by a global U(1) × ℤ2 symmetry (see Fig. 3A). Here, we study its phase diagram. As explained later, the model can be implemented relatively easily in existing ultracold atom setups by coupling multiple double-well systems, which is our main motivation for studying its phase diagram.
(A) We consider the Hamiltonian (Eq. 24) describing a-particles that are minimally coupled to the ℤ2 gauge field
The model. We combine multiple double-well systems (Eq. 12) to a two-leg ladder by introducing tunnelings
Effective Hamiltonian
For a properly designed configuration of lattice gradients and modulations, presented in detail later, we obtain an effective Hamiltonian
Expressions for the amplitude renormalizations λy ∈ ℝ and
For the specific set of driving strengths x = x02 that we encountered already in the double-well problem (see Eq. 16), we find that
is defined as a product over all links 〈i, j〉y on the rungs belonging to the edge ∂p of plaquette p. Hence, for these specific modulation strengths
Symmetries
Now, we discuss the symmetries of the effective Hamiltonian (Eq. 19) at the specific value of the driving strengths x02. In the case of decoupled rungs, i.e., for
with
In summary, the effective model is characterized by the global U(1) symmetry associated with the conservation of the number of a-particles and the global ℤ2 symmetry
Physical constituents
In the following, we will describe the physics of the ladder models using the ingredients of ℤ2 LGTs (see Fig. 1C). The quantized excitations of the ℤ2 lattice gauge field are vortices of the ℤ2 (or Ising) lattice gauge field, so-called visons (13). They are defined on the plaquettes of the ladder: If the plaquette term in Eq. 20 is Bp = 1, there is no vison on p; the presence of an additional ℤ2 flux, Bp = −1, corresponds to a vison excitation on plaquette p. Since the matter field
Quantum phase transitions of matter and gauge fields. We start from the microscopic model in Eq. 19 and simplify it by making a mean field approximation for the renormalized tunneling amplitudes, which depend only weakly on
illustrated in Fig. 3A. Later, by introducing a more sophisticated driving scheme, we will show that this model can also be directly implemented using ultracold atoms. The simpler Hamiltonian (Eq. 24) has identical symmetry properties as Eq. 19. Now, we analyze Eq. 24 by means of the DMRG technique. In the phase diagram, we find at least three distinct phases, resulting from transitions in the gauge and matter field sectors (see Fig. 3B). Here, we describe their main features; for more details, the reader is referred to section S4.
Transition in the matter sector
First, we concentrate on the conceptually simpler phase transition taking place in the charge sector. When the tunneling along the legs is weak,
In the limit l ≃ Lx and Lx → ∞, the observable Op(l) remains finite only in the Mott insulating regime. Our results in Fig. 4A confirm that Op takes large values with a weak size dependence for Na = Lx. On the other hand, when Na/Lx ≠ 1 is slightly increased or decreased, the parity Op immediately becomes significantly smaller and a more pronounced Lx dependence is observed, consistent with a vanishing value in thermodynamic limit as expected for an SF phase.
We consider the Hamiltonian from Eq. 24. (A) In the charge sector, we observe transitions from an SF state, characterized by a vanishing parity correlator Op(Lx/2 → ∞ ) → 0 in the thermodynamic limit, to an insulating rung-Mott state at the commensurate filling Na = Lx, characterized by Op(Lx/2 → ∞ ) > 0 and exponentially decaying correlations. Here, we present exemplary results for
For larger values of
Because of the global U(1) symmetry of the model, an SF-to-Mott transition at Na = Lx in the quasi-1D ladder geometry would be of Berezinskii-Kosterlitz-Thouless (BKT) type (59). Hence, the gap is strongly suppressed and the correlation length is exponentially large close to the transition point, making it impossible to determine conclusively from our numerical results whether the ground state is a gapped Mott state in this regime. For hard-core bosons on a two-leg ladder, this scenario is realized: It has been shown by bosonization that an infinitesimal interleg coupling is sufficient to open up an exponentially small Mott gap (55, 60).
Similar considerations apply at the commensurate fillings Na/Lx = 1/2, 3/2, where previous work on single-component bosons in a two-leg ladder (60) pointed out the possible emergence of an insulating charge density wave (CDW) for large enough
The SF phase observed at incommensurate filling fractions is characterized by a power-law decay of the Green’s function
Transition in the gauge sector
In the gauge sector, described by f-particles, we observe a phase transition when the ratios of the tunnel couplings are tuned. In Fig. 4B, we tune
The observed transition is associated with a spontaneous breaking of the global ℤ2 symmetry (Eq. 23) of the model. The f-particles go from a regime where they are equally distributed between the legs,
The two phases are easily understood in the limiting cases. When
LGTs are characterized by Wilson loops (12). Their closest analogs in our two-leg ladder model are string operators of visons
see Fig. 4D. In the disordered phase (electric field dominates), we found numerically that W(d) → 0 when d → ∞. Sufficiently far from the transition, where finite-size effects are small, our data (see fig. S5) show the expected exponential decay. In the ordered phase (magnetic field dominates), W(d) → W∞ quickly converges to the nonzero value
The qualitatively different behavior of the Wilson loop in the two phases is reminiscent of the phenomenology known from the ubiquitous confinement-deconfinement transitions found in (2 + 1) dimensional LGTs (12, 13, 26): There, visons are gapped in the deconfined phase and the Wilson loop decays only weakly exponentially with a perimeter law; in the confining phase, visons condense and the Wilson loop decays much faster with an exponential area law.
Our numerical results in Fig. 4B indicate that the phase transition in the gauge sector is continuous. Beyond this fact, its nature is difficult to determine. The interactions between the ℤ2 link variables are mediated by the matter field, which has correlations extending over many sites following either a power-law (in the SF regime) or featuring exponential decay with a correlation length ξ ≫ 1 (in the considered Mott-insulating regions). On the one hand, this leads to relatively large finite-size effects in our numerical simulations, which explains the continuous onset of the transition in Fig. 4B. On the other hand, when nonlocal Ising interactions mediated by the gapless matter field compete with the transverse ℤ2 electric field term
Interplay of matter and gauge fields
Last, we discuss the interplay of the observed phase transitions in the gauge and matter sectors. To this end, we find it convenient to consider the phase diagram in the
More interesting physics can happen at the tip of the Mott lobe for commensurate fillings Na = Lx. This corresponds to the hatched regime in Fig. 3B, where we cannot say conclusively if the system is in a gapped Mott phase. To obtain better understanding of the commensurate regime, we first argue that an SF cannot coexist with the ordered phase of the gauge field at commensurate fillings: In this regime, the ℤ2 gauge field acquires a finite expectation value,
To shed more light on this problem, we consider the case when the Mott gap Δ is much larger than the tunneling
Our last argument demonstrates that scenario I in Fig. 3C is realized deep in the Mott phase. In this limit of small
On the other hand, a detailed analysis of the Luttinger-K parameter for larger values of
Implementation: Coupled double-well systems
Now, we describe how the models discussed above, and extensions thereof, can be implemented in state-of-art ultracold atom setups. The double-well system introduced around Eq. 12 constitutes the building block for implementing larger systems with a ℤ2 gauge symmetry, or even genuine ℤ2 LGTs, because it realizes a minimal coupling of the matter field to the gauge field (see Fig. 2C) (12). We start by discussing the two-leg ladder Hamiltonian
Two-leg ladder geometry. The ladder system shown in Fig. 3A can be obtained by combining multiple double wells (Eq. 12) and introducing tunnelings
Multiple double-well systems as described in Fig. 2 are combined to form a two-leg ladder by including hopping elements
As shown in section S2, this setup leads to the effective Hamiltonian (Eq. 19). For the specific set of driving strengths
Simplified model
Now, we discuss a further simplification of the model in Eq. 19, leaving its symmetry group unchanged. We note that, even for the specific choice of the driving strengths
Following (48), we obtain expressions for the restored tunnel couplings along x for an energy offset nω introduced by the Hubbard interactions U = ω between a- and f-particles;
where λx = λ012 ≈ 0.37. A similar approach can be used to make
Realizing a ℤ2 LGT in a 2D square lattice. Now, we present a coupling scheme of double-wells, which results in an effective 2D LGT Hamiltonian with genuine local symmetries, in addition to the global U(1) symmetry associated with a-number conservation. We will derive a model with ℤ2 gauge-invariant minimal coupling terms
Setup
We consider the setup shown in Fig. 6A in a layered 2D optical lattice, which is a particular type of brick-wall lattice. The a-particles tunnel vertically between the layers in the z direction, with coupling matrix element
(A) Multiple double-well systems as described in Fig. 2 are combined in the shown brick-wall lattice. Each of its four layers along the z direction is used to realize one of the four links connecting every lattice site of the 2D square lattice (B) to its four nearest neighbors. The double-well systems are indicated by solid lines (colors), and they are only coupled by tunnelings of a-particles along the z direction, with amplitudes
For the realization of the individual double-well systems, we consider a modulated potential gradient along x and y, seen equally by the matter and gauge fields. The modulation amplitudes
To realize a-particle tunneling along z, which is independent of the ℤ2 gauge fields
Effective Hamiltonian
Combining our results from the previous section, we obtain the effective hopping Hamiltonian
using the same notation as introduced earlier. Here, we treat the z-coordinate nez, with n = 1, …,4, as an internal degree of freedom, while j is a site index in the 2D square lattice; m〈i,j〉 ∈ {1, 2, 3, 4} denotes the z-coordinate corresponding to double well 〈i, j〉. For simplicity, we assumed that
Using the multifrequency driving scheme explained around Eq. 28, a situation where
A simplified effective Hamiltonian, where the internal degrees of freedom are eliminated, can be obtained when
The ground state energy ε2a with two hard-core a-particles tunneling along z in the same supersite is larger than twice the energy εa of a single a-particle, by an amount Ueff, i.e., ε2a = 2εa + Ueff. By solving the one- and two-particle problems exactly, we find
By projecting the Hamiltonian (Eq. 30) to the lowest internal state on every supersite, we arrive at the following simplified model
Here, we distinguish between two sets of links, 〈i,j〉 ∈ E or B, which are realized in layers at the edge n = 1,4 (E) and in the bulk n = 2,3 (B) in the 3D implementation (see Fig. 6A). Because the internal state has different weights ∣ϕ1∣2 ≈ 0.14 and ∣ϕ2∣2 ≈ 0.36, they are associated with different tunneling amplitudes. This complication can be avoided by realizing bare tunnelings of a-particles with different strengths on E- and B-type bonds.
Symmetries
In contrast to the two-leg ladder (Eq. 19), the models in Eqs. 30 and 32 are both characterized by local ℤ2 gauge symmetries. The ℤ2 charge on a supersite is defined as
where the product on the right includes all links 〈j, i〉 connected to site j.
It holds
In addition to the local ℤ2 gauge invariance, the models (Eqs. 30 and 32) have a global U(1) symmetry associated with the conservation of the a-particle number. Very similar Hamiltonians have been studied in the context of strongly correlated electrons, where fractionalized phases with topological order have been identified (35). When the a-particles condense, effective models without the global U(1) symmetry can also be realized. These are in the same symmetry class as Kitaev’s toric code (28).
DISCUSSION
We have presented a general scheme for realizing flux attachment in 2D optical lattices, where one species of atoms becomes a source of magnetic flux for a second species. For a specific set of parameters, we demonstrated that the effective Floquet Hamiltonian describing our system has a ℤ2 gauge structure. This allows us to implement experimentally a dynamical ℤ2 gauge field coupled to matter using ultracold atoms, as we have shown specifically for a double-well setup, two-leg ladders, and in a 2D geometry. Because our scheme naturally goes beyond one spatial dimension, the ℤ2 magnetic field—and the corresponding vison excitations—plays an important role in our theoretical analysis of the ground-state phase diagram. Moreover, the link variables in our system are realized by particle number imbalances on neighboring sites, making experimental implementations of our setup feasible using existing platforms [as described, e.g., in (42, 43, 45)].
Our theoretical analysis of hard-core bosons coupled to ℤ2 link variables on the rungs of a two-leg ladder revealed an SF-to-Mott transition in the charge sector as well as a transition in the gauge sector. The latter is characterized by a spontaneously broken global ℤ2 symmetry, but we argued that it can be considered as a precursor of the confinement-deconfinement transitions, which are ubiquitous in LGTs, high-energy physics, and strongly correlated quantum many-body system. Leveraging the powerful toolbox of quantum gas microscopy, our approach paves the way for new studies of LGTs with full resolution of the quantum mechanical wave function. This is particularly useful for analyzing string (57, 63) and topological (64) order parameters, which are at the heart of LGTs but difficult to access in more conventional settings.
As we have demonstrated, extensions of our LGT setting to 2D systems with local rather than global symmetries are possible. Here, we propose a realistic scheme to implement a genuine ℤ2 LGT with minimal coupling of the matter to the gauge field on all links of a square lattice. On the one hand, this realizes one of the main ingredients of Kitaev’s toric code (28, 65, 66)—a specific version of an LGT coupled to matter, which displays local ℤ2 gauge symmetry and hosts excitations with non-Abelian anyonic statistics. On the other hand, the systems that can be implemented with our technique are reminiscent of models studied in the context of nematic magnets (27, 33, 67) and strongly correlated electron systems (13, 35, 36). Other extensions of our work include studies of more general systems with flux attachment, which are expected to reveal physics related to the formation of composite fermions in the FQH effect.
Another application of our work is the realization of the recently suggested ℤ2 Bose-Hubbard Hamiltonian (68) using ultracold atoms in optical lattices. This model contains ℤ2 link variables on a 1D chain, similar to our case, but includes terms in the Hamiltonian, which explicitly break the local ℤ2 gauge symmetry. In contrast to the models studied in this paper, not only the tunneling phases but also the tunneling amplitudes in the ℤ2 Bose-Hubbard Hamiltonian depend on the ℤ2 link variables. The ℤ2 Bose-Hubbard model features bosonic Peierls transitions (68), which can lead to an interesting interplay of symmetry breaking and symmetry-protected topological order (52, 53).
In terms of experimental implementations, we restricted our discussion in this article to ultracold atom setups. However, other quantum simulation platforms, such as arrays of superconducting qubits (69), provide promising alternatives. Generalizations of our scheme to these systems are straightforward, and a detailed analysis of the feasibility of our proposal in such settings will be devoted to future work.
MATERIALS AND METHODS
Implementing dynamical gauge fields
Here, we describe in detail how synthetic gauge fields with their own quantum dynamics can be realized and implemented using ultracold atoms. We begin by quickly reviewing results for the case of a single particle in a double-well potential, which we use later on to derive the effective Hamiltonian in a many-body system.
Single-particle two-site problem. We consider the following Hamiltonian describing a single particle hopping between sites ∣1〉 and ∣2〉
Here, t > 0 denotes the bare tunnel coupling, which is strongly suppressed by the energy offset ∣Δ2,1 ∣ ≫ t. Tunneling is then restored by a modulation
For resonant shaking, ω = Δ2,1, it has been shown in (47) that the dynamics of Eq. 34 can be described by the following effective Hamiltonian
The amplitude of the restored tunneling is given by
and the complex phase ϕ2,1 is determined directly from the modulating potential
More generally, when the offset Δ2,1 = nω is a positive integer multiple n = 0,1,2,3,4, ... of the driving frequency ω, tunneling can also be restored. As shown by a general formalism in (48), the effective Hamiltonian in this case becomes
For n = 0, the result is independent of the phase ϕ2,1 of the modulation. The tunneling matrix element is renormalized by
The first three Bessel functions, n = 0, 1, 2, are plotted in fig. S1 as a function of x = A2,1/ω.
Last, we consider the case when Δ2,1 = −nω, for a positive integer n = 1, 2, 3, …. In this case, we can rewrite the modulation (Eq. 35) as
i.e., effectively ω → −ω and ϕ2,1 → −ϕ2,1. By applying the results from Eqs. 38 and 39 for the system with −ω, we obtain
The complex phase of the restored hopping in the effective Hamiltonian changes sign, because −ϕ2,1 appears in Eq. 40. In addition, it contains a π phase shift, which takes into account the sign change of the renormalized tunneling matrix element
Multiple driving frequencies. Even more control over the restored tunnel couplings can be gained by using lattice modulations with multiple frequency components. Here, we summarize results for the single-particle two-site problem from above, for the case of driving with frequency components ω and 2ω. To do so, we modify our Hamiltonian in Eq. 35 as
To calculate the effective Hamiltonian, we rewrite the time-dependent Hamiltonian (Eq. 42) in a moving frame by performing a time-dependent unitary transformation realized by the operator (48)
Using the Jacobi-Anger identity
While this effective Hamiltonian is similar to Eq. 38, the amplitude renormalization now involves a product of two Bessel functions
Two-particle two-site problem. Now, we apply the results from the first paragraph (Eqs. 34 to 41)] to the problem of a pair of a- and f-particles in a double-well potential (see Fig. 2). In contrast to the main text, we consider general parameters in our derivation of the effective Hamiltonian. Our starting point is the model in Eqs. 1 to 6 for two sites j1 and j2 = j1 + ey. We assume Va(j1,2) ≡ 0 but introduce a static energy offset Δf = U between the two lattice sites for the f-particles, Vf(j2) = Δf + Vf(j1). Because our analysis is restricted to the subspace with one a-particle and one f-particle, the hard-core constraint assumed in the main text is not required in this case and the statistics of the two species are irrelevant.
The two-site problem has four basis states,
To derive Eqs. 49 to 53, we first consider the effect of the coherent driving
governing the dynamics of
When expressed in terms of the two states
According to Eqs. 38 and 41, the restored tunnel coupling between ∣1〉a and ∣2〉a has a complex phase given by φ = ϕj2, j1 if Δ2,1> 0, i.e., for
Next, we consider the dynamics of the f-particles or, equivalently, the link variable
Because
When expressed in terms of the two states
In the case of f-particles, the energy offset Δ2,1 can only take positive values 0 and 2ω if Δf = U = ω. From Eq. 38, it follows that the restored tunnel coupling between ∣1〉f and ∣2〉f has a complex phase given by θ = 0 if Δ2,1 = 0, i.e., for δna = − 1, and by θ = 2ϕj2, j1 if Δ2,1 = 2ω, i.e., for δna = 1. Expressed in terms of
The magnitudes of the restored tunneling couplings of f-particles in the two-particle Hilbert space depend on the energy offset Δ2,1. In the case when Δ2,1 = 0, i.e., for δna = −1, it becomes
Realizations with ultracold atoms. Next, we discuss realizations of the two-particle two-site problem with ultracold atoms. The proposed implementation needs two distinguishable particles with strong interspecies on-site interaction energy U ≫ ty. The particles occupy a double well with both species-dependent and species-independent on-site potentials. For the species-dependent contribution, a static potential is sufficient, which introduces a tilt Δf = U between neighboring sites for the f-particles but leads to zero tilt for the a-particles. On the other hand, the species-independent contribution must be time-dependent Vω(t) to restore resonant tunneling for both particles.
For ultracold atoms, a cubic array of lattice sites with period ds can be created by three mutually orthogonal standing waves with wavelengths λ = 2ds. When extending this simple cubic lattice along one axes by an additional lattice with twice the period dl = 2ds, a superlattice of the form
The two distinguishable particles can be encoded in different hyperfine sublevels with different magnetic moments, enabling the direct implementation of the static species-dependent potentials by a magnetic field gradient. This is especially appealing for bosonic atoms having a hyperfine sublevel with zero magnetic moment, which directly results in a vanishing, magnetic field–independent tilt for the a-particles in first order. Nevertheless, this is not essential as tilts for the a-particles can be compensated by the present species-independent potentials.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaav7444/DC1
Section S1. Flux attachment in 2D
Section S2. Implementing matter coupled to a ℤ2 gauge field in the two-leg ladder geometry
Section S3. Gauge structure of two-leg ladders
Section S4. Phase transitions of gauge and matter fields
Fig. S1. Renormalized tunneling amplitudes determined by Bessel functions.
Fig. S2. Two-site two-particle problem.
Fig. S3. Flux attachment in a 2D Hofstadter model.
Fig. S4. Derivation of the effective Hamiltonian.
Fig. S5. Wilson loop scaling.
Fig. S6. The Green’s function in the charge sector.
Fig. S7. The Luttinger-K parameter.
Fig. S8. Rung-Mott state at commensurate filling.
Fig. S9. Phase diagram of the ℤ2 LGT on a two-leg ladder for commensurate filling.
Fig. S10. Transition in the gauge sector.
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REFERENCES AND NOTES
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