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arXiv:1709.10409 · Physical review. B./Physical review. B

Haldane phases with ultracold fermionic atoms in double-well optical lattices

Pierre Fromholz, Sylvain Capponi, P. Lecheminant, D. J. Papoular, K. Totsuka

Published Feb 15, 2019• 11 citations
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Abstract

We propose to realize one-dimensional topological phases protected by $\mathrm{SU}(N)$ symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical lattice. We derive a realistic model for this system and investigate it theoretically. Depending on the parity of $N$, two different classes of symmetry-protected topological (SPT) phases are stabilized at half-filling for physical parameters of the model. For even $N$, the celebrated spin-1 Haldane phase and its generalization to $\mathrm{SU}(N)$ are obtained with no local symmetry breaking. In stark contrast, at least for $N=3$, a new class of SPT phases, dubbed chiral Haldane phases, that spontaneously break inversion symmetry, emerges with a twofold ground-state degeneracy. The latter ground states with open-boundary conditions are characterized by different left and right boundary spins, which are related by conjugation. Our results show that topological phases are within close reach of the latest experiments on cold fermions in optical lattices.

DOI
10.1103/physrevb.99.054414

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We propose to realize one-dimensional topological phases protected by SU(N)\mathrm{SU}(N)SU(N) symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical lattice. We derive a realistic model for this system and investigate it theoretically. Depending on the parity of NNN, two different classes of symmetry-protected topological (SPT) phases are stabilized at half-filling for physical parameters of the model. For even NNN, the celebrated spin-1 Haldane phase and its generalization to SU(N)\mathrm{SU}(N)SU(N) are obtained with no local symmetry breaking. In stark contrast, at least for N=3N=3N=3, a new class of SPT phases, dubbed chiral Haldane phases, that spontaneously break inversion symmetry, emerges with a twofold ground-state degeneracy. The latter ground states with open-boundary conditions are characterized by different left and right boundary spins, which are related by conjugation. Our results show that topological phases are within close reach of the latest experiments on cold fermions in optical lattices.
We investigate possible realizations of exotic SU(N)\mathrm{SU}(N)SU(N) symmetry-protected topological (SPT) phases with alkaline-earth cold fermionic atoms loaded into one-dimensional optical lattices. A thorough study of two-orbital generalizations of the standard SU(N)\mathrm{SU}(N)SU(N) Fermi-Hubbard model, directly relevant to recent experiments, is performed. Using state-of-the-art analytical and numerical techniques, we map out the zero-temperature phase diagrams at half-filling and identify several Mott-insulating phases. While some of them are rather conventional (nondegenerate, charge-density wave, or spin-Peierls-like), we also identify, for even NNN, two distinct types of SPT phases: an orbital Haldane phase, analogous to a spin-N/2N/2N/2 Haldane phase, and a topological SU(N)\mathrm{SU}(N)SU(N) phase, which we fully characterize by its entanglement properties. We also propose sets of nonlocal order parameters that characterize the SU(N)\mathrm{SU}(N)SU(N) topological phases found here.
We investigate the existence of symmetry-protected topological phases in one-dimensional alkaline-earth cold fermionic atoms with general half-integer nuclear spin I at halffilling. In this respect, some orbital degrees of freedom are required. They can be introduced by considering either the metastable excited state of alkaline-earth atoms or the p -band of the optical lattice. Using complementary techniques, we show that SU(2) Haldane topological phases are stabilised from these orbital degrees of freedom. On top of these phases, we find the emergence of topological phases with enlarged SU(2 I + 1) symmetry which depend only on the nuclear-spin degrees of freedom. The main physical properties of the latter phases are further studied using a matrix-product state approach. On the one hand, we find that these phases are symmetry-protected topological phases, with respect to inversion symmetry, when which is directly relevant to ytterbium and strontium cold fermions. On the other hand, for the other values of I (=half-odd integer), these topological phases are stabilised only in the presence of exact SU(2 I + 1) symmetry.
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.
We propose that ultracold alkaline-earth-like atoms confined in one-dimensional optical lattice can realize a Kondo lattice model which hosts a symmetry-protected topological (SPT) phase and an associated quantum phase transition in a controllable manner. The symmetry protection of the phase transition is discussed from two different viewpoints: topological properties related to spatial patterns of Kondo singlets and symmetry eigenvalues of the spin states. We uncover the role of various symmetries in the phase diagram of this system by combining a weak-coupling approach by Abelian bosonization and strong-coupling pictures of ground states. Furthermore, we show that the bosonization approach correctly describes a crossover from a fermionic SPT phase to a bosonic SPT phase and an associated change of protecting symmetries as the charge degrees of freedom are frozen by the Hubbard repulsion.
It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2NNN) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold M\mathcal{M}M. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for N>1N>1N>1, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2NNN) spin symmetry, or its subgroup SU(NNN)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2NNN) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.
In this paper, we investigate topological properties of the ground state of the SU(N) Heisenberg chain, which is argued to be relevant to the Mott-insulating phase of alkaline-earth cold fermions in a one-dimensional optical lattice. By calculating the entanglement spectrum, we show that the ground state is in one of the topological phases protected by SU(N) symmetry. We then discuss an alternative characterization of it with non-local string order parameters. We also consider how the reduction of the protecting symmetry affects the topological phase paying particular attention to the entanglement spectrum.