Engineering Ising-XY spin-models in a triangular lattice using tunable artificial gauge fields

Abstract

Emulation of gauge fields for ultracold atoms provides access to a class of exotic states arising in strong magnetic fields. Here we report on the experimental realisation of tunable staggered gauge fields in a periodically driven triangular lattice. For maximal staggered magnetic fluxes, the doubly degenerate superfluid ground state breaks both a discrete Z 2 (Ising) symmetry and a continuous U (1) symmetry. By measuring an Ising order parameter, we observe a thermally driven phase transition from an ordered antiferromagnetic to an unordered paramagnetic state and textbook-like magnetisation curves. Both the experimental and theoretical analysis of the coherence properties of the ultracold gas demonstrate the strong influence of the Z 2 symmetry onto the condensed phase.Phase transitions in systems with combined continuous and discrete symmetries are fundamentally different from their purely continuous and discrete counterparts. The interplay between different types of excitations in the various degrees of freedom can lead to a complex behaviour and coupling of the associated order parameters [1][2][3][4][5]. A paradigm example is the fully frustrated XY model on a triangular lattice. It combines vector spin-type symmetries with discrete chiral degrees of freedom, which result in the famous spin-chirality coupling at low temperatures [6]. However, experimental studies in solid-state systems are challenging in view of implementing and isolating an XY model Hamiltonian [7][8][9].Ultracold bosonic quantum gases in optical lattices, on the other hand, constitute a highly versatile system with an extraordinary degree of control [10,11]. In particular, the recent experimental realisations of artificial gauge potentials for bulk [12][13][14][15] and optical lattice systems [16][17][18][19] allow for the investigation of new physical regimes, not realisable in condensed matter systems.Here, we demonstrate the realisation of a system with combined U (1) and Z 2 symmetries using ultracold atoms submitted to artificial gauge fields. Our experimental setup consists of an ultracold gas of 87 Rb atoms held in a two-dimensional triangular lattice [20] (see Fig. 1a). At each lattice site j with particle number N j , the weakly interacting superfluid gas can be described by the local order parameter a j = N j e iϕj . As a central aspect, the local phases ϕ j are mapped onto classical XY spins s j = (cos ϕ j , sin ϕ j ), where the tunneling matrix elements between neighbouring lattice sites correspond to the spinspin coupling parameters. Such classical spins possess a continuous degree of freedom. In presence of a long-range order, analogous to the onset of Bose-Einstein condensation (BEC), the order parameter assumes an arbitrary, but fixed phase, thus breaking the continuous U (1) symmetry [21].Beyond that, we experimentally engineer strong staggered gauge fields, which generate an additional discrete Z 2 symmetry in our system. The resulting magnetic flux induces cyclotron-like mass currents around each plaquette. The two poss...

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

“…The HH model arises after time averaging the Floquet Hamiltonian, but it is an open question to what extent finite interactions and micromotion lead to transitions between Floquet modes and therefore heating [21][22][23][24]. Bose-Einstein condensation has been achieved in staggered flux configurations [15,25] and in small ladder systems [26,27] further highlighting its noted absence in the uniform field configuration.In this article, we report the first observation of BoseEinstein condensation in Hofstadter's butterfly. The presence of a superfluid state in the HH lattice allows us to analyze the symmetry of the periodic wavefunction by self-diffraction of coherent matter waves during ballistic expansion -analogous to a von Laue x-ray diffraction pattern, which reveals the symmetry of a lattice.…”

“…The HH model arises after time averaging the Floquet Hamiltonian, but it is an open question to what extent finite interactions and micromotion lead to transitions between Floquet modes and therefore heating [21][22][23][24]. Bose-Einstein condensation has been achieved in staggered flux configurations [15,25] and in small ladder systems [26,27] further highlighting its noted absence in the uniform field configuration.…”

“…For neutral ultracold atoms, new methods have been developed to create synthetic gauge fields. Forces analogous to the Lorentz force on electons are engineered either through the Coriolis force in rotating systems [7][8][9], by phase imprinting via photon recoil [10][11][12][13][14], or lattice shaking [15,16]. Much of the research with ultracold atoms has focused on the paradigmatic HarperHofstadter (HH) Hamiltonian, which describes particles in a crystal lattice subject to a strong homogeneous magnetic field [17][18][19].…”

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

“…Treating interactions in such a non-perturbative way is difficult in periodically-driven systems [5][6][7][8][9][10], which have received unprecedented attention following the realisation of dynamical localisation [11][12][13][14][15], artificial gauge fields [16][17][18][19][20][21][22], models with topological [23][24][25][26][27][28] and statedependent [29] bands, and spin-orbit coupling [30,31]. In this paper, we consider strongly-interacting periodicallydriven systems and show how the SWT can be extended to derive effective static Hamiltonians of non-equilibrium setups.…”

Topological phases of lattice bosons with a dynamical gauge field

“…The synthetic implementation of this mechanism in cold atomic systems has been envisaged since the early days of quantum gases [1][2][3][4][5][6][7], and has been realized successfully during recent years [8][9][10][11][12][13]. Current quantum simulations with artificial gauge potentials are exploring the variety of interesting physics related to background gauge fields: spin liquid phases [14], topological phases evidenced by nonzero Chern numbers [15], or quantum Hall phases with edge currents [16,17]. A long-term goal is the simulation of quantum electromagnetism or chromodynamics, that is, of models where matter interacts with dynamical fields, as described in Refs.…”