Research ArticlePHYSICS

Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to ℤ2 lattice gauge theories

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Science Advances  11 Oct 2019:
Vol. 5, no. 10, eaav7444
DOI: 10.1126/sciadv.aav7444
  • Fig. 1 Flux-attachment and dynamical gauge fields with 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 a^ has a ℤ2 charge given by the parity of its occupation numbers n^a. It couples to the ℤ2 gauge field τ^i,jz, defined as the number imbalance of the f-particles between different ends of a link. When ∣ta∣≪∣tf∣, the ground state is dominated by tunneling of the f-particles, realizing that eigenstates of the ℤ2 electric field τ^i,jx delocalized over the link. In the opposite limit, ∣ta∣≫∣tf∣, the tunneling dynamics of the a-particles prevails and the system realizes eigenstates of the ℤ2 magnetic field B^p, defined as a product of the gauge field τ^z over all links ℓ ∈ ∂p along the edge of a plaquette p. The ℤ2 magnetic field introduces Aharonov-Bohm phases for the matter field, which are 0 (π) when the f particles occupy the same (different) leg of the ladder, i.e., if Bp = 1 (Bp = −1). The quantized excitations of the dynamical gauge field correspond to ℤ2 vortices of the Ising gauge field, so-called visons.

  • Fig. 2 2 LGT in a two-well system.

    (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 a^ to the link variable defined by the f-particles. The action of this term in the effective Hamiltonian acting on a basis state (left) is illustrated. The strength of the ℤ2 electric field is indicated by the thickness of the blue line connecting the two sites of the double-well system. The minimal coupling term is the common building block for realizing larger systems with a ℤ2 gauge structure. (D) These systems are characterized by a symmetry G^j associated with each lattice site j. Here, G^j commutes with the Hamiltonian and consists of the product of the ℤ2 charge, Q^j=(1)n^ja, and all electric field lines—for which τx = −1—emanating from a volume around site j.

  • Fig. 3 Coupling matter to a ℤ2 gauge field in a two-leg ladder.

    (A) We consider the Hamiltonian (Eq. 24) describing a-particles that are minimally coupled to the ℤ2 gauge field τ^i,jyz on the rungs of a two-leg ladder. (B) The phase diagram, obtained by DMRG simulations at tyf/txa=0.54, contains an SF-to-Mott transition in the charge sector at a commensurate density of the matter field, Na = Lx. In addition, we find a transition in the gauge sector, from an ordered region with a broken global ℤ2 symmetry where the ℤ2 magnetic field dominates and the vison excitations of the gauge field are gapped (red) to a disordered regime where the ℤ2 electric field is dominant and visons are strongly fluctuating in a condensed state (blue). Along the hatched lines at commensurate fillings Na/Lx = 1/2,1, 3/2, insulating CDW states could exist, but conclusive numerical results are difficult to obtain. (C) The conjectured schematic phase diagram of Eq. 24 is shown in the μtyf plane, where μ denotes the chemical potential for a^ particles and 2tyf corresponds to the energy cost per ℤ2 electric field line along a rung. Two scenarios are realized in different parameter regimes: In scenario I, the interplay of gauge and matter fields prevents a fully disordered Mott phase, whereas the latter exists in scenario II. The behavior in scenario I resembles the phase diagram of the more general 2D ℤ2 LGT (13, 2628) sketched in (D). In our DMRG simulations here, as well as in the following figures, we keep up to 1400 DMRG states with five finite-size sweeps; the relative error on the energies is kept smaller than 10−7.

  • Fig. 4 Characterizing phase transitions of matter coupled to a ℤ2 gauge field in a two-leg ladder.

    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 tya/txa=3 and tyf/txa=0.54. (B) In the gauge sector, we find a transition from a disordered phase, where the ℤ2 electric field dominates, to a phase where the ℤ2 magnetic field dominates. In the second case, the order parameter τ^i,jyz0 corresponds to a spontaneously broken global ℤ2 symmetry (23). In the two phases, the corresponding vison excitations of the ℤ2 gauge field (C) have different characteristics. The numerical results in (A) [respectively (B)] are obtained by considering periodic boundary conditions (respectively Lx = 96 rungs with open boundaries). (D) Analogs of Wilson loops W^(d) in the two-leg ladder are string operators of visons.

  • Fig. 5 Implementing matter-gauge field coupling in a two-leg ladder.

    Multiple double-well systems as described in Fig. 2 are combined to form a two-leg ladder by including hopping elements txa of the a-particles along the x direction. Coherent tunneling is first suppressed by strong interspecies Hubbard interactions U and static potential gradients: Δxa=U for a-particles along x, and Δyf=U for f-particles along y. These gradients are indicated by triangles whose colors refer to the respective atomic species. The tunnel couplings are restored by a resonant lattice shaking with frequency ω = U, realized by a modulated potential gradient Vω(j,t)=(jxVωx+jyVωy)cos (ωt) seen by both species. The modulated gradients are indicated by light-colored triangles. We assume that each rung is occupied by exactly one f-particle, which can thus be described by a link variable, while the number Na of a-particles is freely tunable. As shown in section S2, the special choice for the driving strengths Vωx/ω=Vωy/ω=x02 leads to an effective Hamiltonian with matter coupled to ℤ2 lattice gauge fields on the rungs. The gradient Δxa=U guarantees that the a-particles pick up only trivial phases φ^x=0 while tunneling along the legs of the ladder. Hence, the Aharonov-Bohm phases (red) associated with the matter field become 0, or π corresponding to a vison excitation. They are determined by the plaquette terms B^p defined in Eq. 20, reflecting the configuration of f-particles.

  • Fig. 6 Realizing ℤ2 LGT coupled to matter in 2D.

    (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 tza. The required lattice gradients (their modulations) are indicated by (light) colored triangles. (B) The restored hopping Hamiltonian H^2DLGT in the 2D lattice has local symmetries G^j associated with all lattice sites j, i.e., [H^2DLGT,G^j]=0.

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.

    References (7074)

  • Supplementary Materials

    This PDF file includes:

    • 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.
    • References (7074)

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