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arXiv:1606.04297 · Physical review. B./Physical review. B

Haldane insulator protected by reflection symmetry in the doped Hubbard model on the three-legged ladder

H. L. Nourse, Ian P. McCulloch, C. Janani, Benjamin James Powell

Published Dec 19, 2016• 17 citations
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Abstract

We demonstrate the existence of an insulating phase in the three-legged Hubbard ladder at two-thirds filling. In this phase chargons are bound because the physics within a unit cell favors the formation of triplets. The resultant moments lead to a ground state in the Haldane phase, a symmetry protected topological state of matter. In this purely fermionic model, reflection is protecting but time-reversal and dihedral symmetries are not, in contrast to spin models.

DOI
10.1103/physrevb.94.214418

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We demonstrate the existence of an insulating phase in the three-legged Hubbard ladder at two-thirds filling. In this phase chargons are bound because the physics within a unit cell favors the formation of triplets. The resultant moments lead to a ground state in the Haldane phase, a symmetry protected topological state of matter. In this purely fermionic model, reflection is protecting but time-reversal and dihedral symmetries are not, in contrast to spin models.
Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective SU(N)\mathrm{SU}(N)SU(N) group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with N>2N>2N>2. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of SU(N)\mathrm{SU}(N)SU(N). We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for N=3N=3N=3 and 4 to show that all the possible symmetry-protected topological phases with projective SU(N)\mathrm{SU}(N)SU(N) symmetry appear in this simple ladder model.
Anfuso and Rosch [Phys. Rev. B 75, 144420 (2007)] showed that the ``topological'' Haldane phase in a fermionic spin-1/21/21/2 ladder can be continuously deformed into a ``trivial'' phase without explicitly breaking symmetries when local charge fluctuations are taken into account. Within the framework of symmetry-protected topological phases, we revisit the model and demonstrate how the Haldane phase can be adiabatically connected to a trivial phase due to charge fluctuations. Furthermore, we show that the Haldane phase remains stable as long as the system is symmetric under particular reflection symmetries.
Recent experiments on ultracold dipoles in optical lattices open exciting possibilities for the quantum simulation of extended Hubbard models. When considered in one dimension, these models present at unit filling a particularly interesting ground-state physics, including a symmetry-protected topological phase known as Haldane insulator. We show that the tail of the dipolar interaction beyond nearest-neighbors, which may be tailored by means of the transversal confinement, does not only modify quantitatively the Haldane insulator regime and lead to density waves of larger periods, but results as well in unexpected insulating phases. These insulating phases may be topological or topologically trivial, and are characterized by peculiar correlations of the site occupations. These phases may be realized and observed in state-of-the-art experiments.
Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U(1) charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from strong interactions.
Strong interactions can give rise to new fermionic symmetry-protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U(1)U(1)U(1) charge conservation and time-reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the noninteracting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry-protected topological phase resulting from strong interactions.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.