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arXiv:1302.6136 · Physical Review B

How hidden orders generate gaps in one-dimensional fermionic systems

Luca Barbiero, Arianna Montorsi, M. Roncaglia

Published Jul 8, 2013• 47 citations
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Abstract

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.

DOI
10.1103/physrevb.88.035109

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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 opening of a charge gap driven by interaction is a fingerprint of the transition to a Mott insulating phase. In strongly correlated low-dimensional quantum systems, it can be associated to the ordering of hidden non-local operators. For Fermionic 1D models, in the presence of spin–charge separation and short-ranged interaction, a bosonization analysis proves that such operators are the parity and/or string charge operators. In fact, a finite fractional non-local parity charge order is also capable of characterizing some two-dimensional Mott insulators, in both the Fermionic and the bosonic cases. When string charge order takes place in 1D, degenerate edge modes with fractional charge appear, peculiar of a topological insulator. In this article, we review the above framework, and we test it to investigate through density-matrix-renormalization-group (DMRG) numerical analysis the robustness of both hidden orders at half-filling in the 1D Fermionic Hubbard model extended with long range density-density interaction. The preliminary results obtained at finite size including several neighbors in the case of dipolar, screened and unscreened repulsive Coulomb interactions, confirm the phase diagram of the standard extended Hubbard model. Besides the trivial Mott phase, the bond ordered and charge density wave insulating phases are also not destroyed by longer ranged interaction, and still manifest hidden non-local orders.
We investigate the phase diagram of spinless bosons with long range (~ 1/r^3) repulsive interactions, relevant to ultracold polarized atoms or molecules, using DMRG. Between the two conventional insulating phases, the Mott and density wave phases, we find a new phase possessing hidden order revealed by non local string correlations analogous to those characterizing the Haldane gapped phase of integer spin chains. We develop a mean field theory that describes the low energy excitations in all three insulating phases. This is used to calculate the absorption spectrum due to oscillatory lattice modulation. We predict a sharp resonance in the spectrum due to a collective excitation of the new phase that would provide clear evidence for the existence of this phase.
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.
Ultracold fermions in one-dimensional, spin-dependent nonoverlapping optical lattices are described by a nonstandard Hubbard model with next-nearest-neighbor correlated hopping. In the limit of a kinetically constraining value of the correlated hopping equal to the normal hopping, we map the invariant subspaces of the Hamiltonian exactly to free spinless fermion chains of varying lengths. As a result, the system exactly manifests spin-charge separation and we obtain the system properties for arbitrary filling: ground state collective order characterized by a spin gap, which can be ascribed to an unconventional critical hole superconductor associated with finite long range nonlocal string order. We study the system numerically away from the integrable point and show the persistence of both long range string order and spin gap for appropriate parameters as well as a transition to a ferromagnetic state.
We characterize the Mott-insulator and Luther-Emery phases of the 1D Hubbard model through correlators that measure the parity of spin and charge strings along the chain. These nonlocal quantities order in the corresponding gapped phases and vanish at the critical point U(c)=0, thus configuring as hidden order parameters. The Mott insulator consists of bound doublon-holon pairs, which in the Luther-Emery phase turn into electron pairs with opposite spins, both unbinding at U(c). The behavior of the parity correlators is captured by an effective free spinless fermion model.
Topological phases, like the Haldane phase in spin-1 chains, defy characterization through local order parameters. Instead, nonlocal string order parameters can be employed to reveal their hidden order. Similar diluted magnetic correlations appear in doped one-dimensional lattice systems owing to the phenomenon of spin-charge separation. Here we report on the direct observation of such hidden magnetic correlations via quantum gas microscopy of hole-doped ultracold Fermi-Hubbard chains. The measurement of nonlocal spin-density correlation functions reveals a hidden finite-range antiferromagnetic order, a direct consequence of spin-charge separation. Our technique, which measures nonlocal order directly, can be readily extended to higher dimensions to study the complex interplay between magnetic order and density fluctuations.