We calculate the properties of a one-dimensional ${Z}_{2}$ lattice gauge theory in different Gauss-law sectors, corresponding to different configurations of static charges, set by the orientations of the gauge spins. Importantly, in quantum simulator experiments these sectors can be accessed without adding any additional physical particles or changing the Hamiltonian: the Gauss-law sectors are simply set by the initial conditions. We study the interplay between conservation laws and interactions when the static charges are chosen to form periodic patterns. We classify the different Gauss-law sectors and use the density matrix renormalization group to calculate the ground-state compressibility, density profiles, charge-density-wave order parameters, and single-particle correlation functions as a function of matter density. We find confined and deconfined phases, charge density waves, correlated insulators, and supersolids.
We calculate the properties of a one-dimensional Z2 lattice gauge theory in different Gauss-law sectors, corresponding to different configurations of static charges, set by the orientations of the gauge spins. Importantly, in quantum simulator experiments these sectors can be accessed without adding any additional physical particles or changing the Hamiltonian: the Gauss-law sectors are simply set by the initial conditions. We study the interplay between conservation laws and interactions when the static charges are chosen to form periodic patterns. We classify the different Gauss-law sectors and use the density matrix renormalization group to calculate the ground-state compressibility, density profiles, charge-density-wave order parameters, and single-particle correlation functions as a function of matter density. We find confined and deconfined phases, charge density waves, correlated insulators, and supersolids.
We explore a Z2 Hamiltonian lattice gauge theory in one spatial dimension with a coupling h, without imposing any Gauss' law constraint. We show that in our model h=0 is a free deconfined quantum critical point containing massless fermions where all Gauss' law sectors are equivalent. The coupling h is a relevant perturbation of this critical point and fermions become massive due to confinement and chiral symmetry breaking. To study the emergent Gauss' law sectors at low temperatures in this massive phase we use a quantum Monte Carlo method that samples configurations of the partition function written in a basis in which local conserved charges are diagonal. We find that two Gauss' law sectors, related by particle-hole symmetry, emerge naturally. When the system is doped with an extra particle, many more Gauss's law sectors related by translation invariance emerge. Using results in the range 0.01<h≤0.15 we find that three different mass scales of the model behave like hp where p≈0.579.
Phase structure of the one-dimensional Z2 lattice gauge theory with second nearest-neighbor interactions
We investigate the ground-state phase diagram of a one-dimensional Z2 lattice gauge theory (LGT) model with hard-core bosons at half-filling, extending previous studies by including second nearest-neighbor (2NN) interactions. Using matrix product state techniques within the density matrix renormalization group, we compute charge gap, static structure factor, pair-pair correlation functions, and entanglement entropy for various interaction strengths and field parameters. We analyze two representative neatest-neighbor interaction strengths (V1) that correspond to the Luttinger liquid (LL) and Mott insulator (MI) phases in the absence of the 2NN interactions. We introduce the 2NN coupling V2 and investigate its impact on the system. Our results reveal very rich behavior. As the 2NN repulsion increases, in the case of small V1, we observe a direct transition from the LL phase to a charge-ordered insulator (COI) phase with four-site ordering pattern, whereas for large V1, we observe a transition from the MI phase with two-site ordering pattern (previously found with only V1 included), going through an intermediate LL region, and finally reaching the COI regime. Additionally, the inclusion of 2NN interactions enhances charge order and suppresses pair coherence, evidenced by sharp peaks in the structure factor and rapid decay in pair-pair correlators. Our work extends the well-studied phase structure of 1D Z2 LGT models and demonstrates the interplay between gauge fields, confinement, and extended interactions.
Lattice gauge theories are a powerful language to theoretically describe a variety of strongly correlated systems, including frustrated magnets, high-Tc superconductors, and topological phases. However, in many cases gauge fields couple to gapless matter degrees of freedom, and such theories become notoriously difficult to analyze quantitatively. In this paper we study several examples of Z2 lattice gauge theories with gapless fermions at finite density, in one and two spatial dimensions, that are either exactly soluble or whose solution reduces to that of a known problem. We consider complex fermions (spinless and spinful) as well as Majorana fermions and study both theories where Gauss' law is strictly imposed and those where all background charge sectors are kept in the physical Hilbert space. We use a combination of duality mappings and the Z2 slave-spin representation to map our gauge theories to models of gauge-invariant fermions that are either free, or with on-site interactions of the Hubbard or Falicov-Kimball type that are amenable to further analysis. In 1D, the phase diagrams of these theories include free-fermion metals, insulators, and superconductors, Luttinger liquids, and correlated insulators. In 2D, we find a variety of gapped and gapless phases, the latter including uniform and spatially modulated flux phases featuring emergent Dirac fermions, some violating Luttinger's theorem.
Confinement is an ubiquitous phenomenon when matter couples to gauge fields, which manifests itself in a linear string potential between two static charges. Although gauge fields can be integrated out in one dimension, they can mediate nonlocal interactions which in turn influence the paradigmatic Luttinger liquid properties. However, when the charges become dynamical and their densities finite, understanding confinement becomes challenging. Here we show that confinement in 1D Z_{2} lattice gauge theories, with dynamical matter fields and arbitrary densities, is related to translational symmetry breaking in a nonlocal basis. The exact transformation to this string-length basis leads us to an exact mapping of Luttinger parameters reminiscent of a Luther-Emery rescaling. We include the effects of local, but beyond contact, interactions between the matter particles, and show that confined mesons can form a Mott-insulating state when the deconfined charges cannot. While the transition to the Mott state cannot be detected in the Green's function of the charges, we show that the metallic state is characterized by hidden off-diagonal quasi-long-range order. Our predictions provide new insights to the physics of confinement of dynamical charges, and can be experimentally addressed in Rydberg-dressed quantum gases in optical lattices.
Exploring confinement transitions in Z2 lattice gauge theories with dipolar atoms beyond one dimension
Confinement of particles into bound states is a phenomenon spanning from high-energy to condensed matter physics, which can be studied in the framework of lattice gauge theories (LGTs). Achieving a comprehensive understanding of confinement continues to pose a major challenge, in particular at finite matter density and in the presence of strong quantum fluctuations. State-of-the-art quantum simulators constitute a promising platform to address this problem. Here we study confinement in coupled chains of Z2 LGTs coupled to matter fields, that can be mapped to a mixed-dimensional (mixD) XXZ model. We perform large-scale numerical matrix-product state calculations to obtain the phase diagram of this model, in which we uncover striped phases formed by the Z2 charges that can be melted at finite temperature or by increasing the tunneling rate. To explore this setting experimentally, we use our quantum simulator constituted by erbium atoms with dipolar interactions in a quantum gas microscope, and observe the predicted melting of a stripe phase by increasing the particle tunneling rate. Our explorative experimental studies of thermal deconfinement of Z2 charges motivate our further theoretical study of the mixD Z2 LGT, in which we predict a confined meson gas at finite temperature and low magnetization where thermal fluctuations destroy stripes but enable spontaneous commensurate spin order. Overall, we demonstrate that our platform can be used to study confinement in Z2 LGTs coupled to matter fields, including long-range interactions and beyond one dimension, paving the way for future research of confinement in the quantum many-body regime.
Two-dimensional Z2 lattice gauge theory on a near-term quantum simulator: variational quantum optimization, confinement, and topological order
We propose an implementation of a two-dimensional Z2 lattice gauge theory model on a shallow quantum circuit, involving a number of single and two-qubits gates comparable to what can be achieved with present-day and near-future technologies. The ground state preparation is numerically analyzed on a small lattice with a variational quantum algorithm, which requires a small number of parameters to reach high fidelities and can be efficiently scaled up on larger systems. Despite the reduced size of the lattice we consider, a transition between confined and deconfined regimes can be detected by measuring expectation values of Wilson loop operators or the topological entropy. Moreover, if periodic boundary conditions are implemented, the same optimal solution is transferable among all four different topological sectors, without any need for further optimization on the variational parameters. Our work shows that variational quantum algorithms provide a useful technique to be added in the growing toolbox for digital simulations of lattice gauge theories.