The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.

We demonstrate that hidden long-range order is always present in the gapped phases of interacting fermionic systems on one dimensional lattices. It is captured by correlation functions of appropriate nonlocal charge and/or spin operators, which remain asymptotically finite. The corresponding microscopic orders are classified. The results are confirmed by DMRG numerical simulation of the phase diagram of the extended Hubbard model, and of a Haldane insulator phase.

The existence of a gapped chiral spin liquid has been recently suggested in the vicinity of the metal-insulator transition of the Hubbard model on the triangular lattice, by intensive density-matrix renormalization group (DMRG) simulations [A. Szasz, J. Motruk, M. P. Zaletel, and J. E. Moore, Phys. Rev. X 10, 021042 (2020)]. Here, we report the results obtained within the variational Monte Carlo technique based upon Jastrow-Slater wave functions, implemented with backflow correlations. As in DMRG calculations, we consider $N$-leg cylinders. For $N=4$ and in the presence of a next-nearest-neighbor hopping, a chiral spin liquid emerges between the metal and the insulator with magnetic quasi-long-range order. Within our approach, the chiral state is gapped and breaks the reflection symmetry. By contrast, for both $N=5$ and 6, the chiral spin liquid is not the state with the lowest variational energy: in the former case, a nematic spin liquid is found in the entire insulating regime, while for the less frustrated case with $N=6$ the results are very similar to that obtained on two-dimensional clusters [L. F. Tocchio, A. Montorsi, and F. Becca, Phys. Rev. B 102, 115150 (2020)], with an antiferromagnetic phase close to the metal-insulator transition and a nematic spin liquid in the strong-coupling regime.

The Mott insulator is characterized by having small deviations around the (integer) average particle density $n$, with pairs with $n-1$ and $n+1$ particles forming bound states. In one dimension, the effect is captured by a nonzero value of a nonlocal “string” of parities, which instead vanishes in the superfluid phase where density fluctuations are large. Here, we investigate the interaction induced transition from the superfluid to the Mott insulator, in the paradigmatic Bose Hubbard model at $n=1$. By means of quantum Monte Carlo simulations and finite size scaling analysis on $L\times M$ ladders, we explore the behavior of “brane” parity operators from one dimension (i.e., $M=1$ and $L\to \infty$) to two dimensions (i.e., $M\to \infty$ and $L\to \infty$). We confirm the conjecture that, adopting a standard definition, their average value decays to zero in two dimensions also in the insulating phase, evaluating the scaling factor of the “perimeter law” [S. P. Rath et al., Ann. Phys. (Berlin) 334, 256 (2013)]. Upon introducing a further phase in the brane parity, we show that its expectation value becomes nonzero in the insulator, while still vanishing at the transition to the superfluid phase. These quantities are directly accessible to experimental measures, thus providing an insightful signature of the Mott insulator.

The interplay between antiferromagnetic interaction and hole motion is capable of inducing conducting Haldane phases with topological features described by a finite nonlocal string order parameter. Here we show that these states of matter are captured by the one-dimensional $t\text{−}{J}_z$ model, which can be experimentally realized with dressed Rydberg atoms trapped onto a one-dimensional optical lattice. In the sector with vanishing total magnetization, exact calculations associated with the bosonization technique allow us to predict that both metallic and superconducting topological Haldane states can be achieved. With the addition of an appropriate magnetic field, the system enters a domain-wall structure with finite total magnetization. In this regime, the conducting Haldane states are confined in domains separated by regions where a fully polarized Luttinger liquid occurs. A procedure to dynamically stabilize such topological phases starting from a confined Ising state is also described.

We investigate entanglement properties at quantum phase transitions (QPTs) of a one-dimensional integrable extended Hubbard model in the momentum-space representation. Two elementary subsystems are recognized: the single mode of an electron and the pair of modes (electrons coupled through the $\eta$-pairing mechanism). We first detect the two or multipartite nature of each quantum phase transition by a comparative study of the singularities of Von Neumann entropy and quantum mutual information. We establish the existing relations between the quantum correlations in the momentum representation and the relevant (superconducting) ones exhibited in the complementary picture: the direct-lattice representation. We then investigate the presence of multipartite entanglement in momentum space by means of a generalization of the Meyer-Wallach measure and we relate its behavior to the nature of the various QPTs.