Author Correction: Direct observation of incommensurate magnetism in Hubbard chains.

In this Letter, the affiliation for Christian Gross should have been 'Max-Planck-Institut für Quantenoptik, Garching, Germany' instead of 'Fakultät für Physik, Ludwig-Maximilians-Universität, Munich, Germany'; this has been corrected online.

Direct observation of incommensurate magnetism in Hubbard chains.

The interplay between magnetism and doping is at the origin of exotic strongly correlated electronic phases and can lead to novel forms of magnetic ordering. One example is the emergence of incommensurate spin-density waves, which have wavevectors that do not belong to the reciprocal lattice. In one dimension this effect is a hallmark of Luttinger liquid theory, which also describes the low-energy physics of the Hubbard model1. Here we use a quantum simulator that uses ultracold fermions in an optical lattice2-8 to directly observe such incommensurate spin correlations in doped and spin-imbalanced Hubbard chains using fully spin- and density-resolved quantum gas microscopy. Doping is found to induce a linear change in the spin-density wavevector, in excellent agreement with predictions from Luttinger theory. For non-zero polarization we observe a reduction in the wavevector with magnetization, as expected from the antiferromagnetic Heisenberg model in a magnetic field. We trace the microscopic-scale origin of these incommensurate correlations to holes, doublons (double occupancies) and excess spins, which act as delocalized domain walls for the antiferromagnetic order. In addition, by inducing interchain coupling we observe fundamentally different spin correlations around doublons and suppression of incommensurate magnetism at finite (low) temperature in the two-dimensional regime9. Our results demonstrate how access to the full counting statistics of all local degrees of freedom can be used to study fundamental phenomena in strongly correlated many-body physics.

Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains.

The repulsive Hubbard Hamiltonian is one of the foundational models describing strongly correlated electrons and is believed to capture essential aspects of high-temperature superconductivity. Ultracold fermions in optical lattices allow for the simulation of the Hubbard Hamiltonian with control over kinetic energy, interactions, and doping. A great challenge is to reach the required low entropy and to observe antiferromagnetic spin correlations beyond nearest neighbors, for which quantum gas microscopes are ideal. Here, we report on the direct, single-site resolved detection of antiferromagnetic correlations extending up to three sites in spin-1/2 Hubbard chains, which requires entropies per particle well below s* = ln(2). The simultaneous detection of spin and density opens the route toward the study of the interplay between magnetic ordering and doping in various dimensions.

Finite-size scaling at the first-order quantum transitions of quantum Potts chains.

We investigate finite-size effects at first-order quantum transitions. For this purpose we consider the one-dimensional q-state quantum Potts chain, in particular with q=10, which undergoes a first-order transition, separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground state. In agreement with the general theory, around the transition the low-energy properties show finite-size scaling with respect to appropriate scaling variables. Their size dependence is particularly sensitive to boundary conditions, which is a specific feature of first-order quantum transitions. Finally, we also discuss the finite-size behavior of the q-state Potts model (q≥2) at the first-order transitions driven by a parallel magnetic field, occurring in the ferromagnetic phase.

Scaling phenomena driven by inhomogeneous conditions at first-order quantum transitions.

We investigate the effects of smooth inhomogeneities at first-order quantum transitions (FOQTs), such as those arising in the presence of a space-dependent external field, which smooths out the discontinuities of the low-energy properties at the transition. We argue that a universal scaling behavior emerges in the space transition region close to the point in which the external field takes the value for which the homogeneous system undergoes the FOQT. We verify the general theory in two model systems. We consider the quantum Ising chain in the ferromagnetic phase and the q-state Potts chain for q=10, investigating the scaling behavior which arises in the presence of an additional inhomogeneous parallel and transverse magnetic field, respectively. Numerical results are in full agreement with the general theory.

Finite-size scaling at first-order quantum transitions.

We study finite-size effects at first-order quantum transitions (FOQTs). We show that the low-energy properties show a finite-size scaling (FSS) behavior, the relevant scaling variable being the ratio of the energy associated with the perturbation driving the transition and the finite-size energy gap at the FOQT point. The size dependence of the scaling variable is therefore essentially determined by the size dependence of the gap at the transition, which in turn depends on the boundary conditions. Our results have broad validity and, in particular, apply to any FOQT characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. In this case, a phenomenological two-level theory provides exact expressions for the scaling functions. Numerical results for the quantum Ising chain in transverse and parallel magnetic fields support the FSS Ansätzes.