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arXiv:1410.2974 · Physical Review B

Phase diagrams of one-dimensional half-filled two-orbital SU(N) cold fermion systems

V. Bois, Sylvain Capponi, P. Lecheminant, Marion Moliner, K. Totsuka

Published Feb 23, 2015• 39 citations
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Abstract

We investigate possible realizations of exotic $\mathrm{SU}(N)$ 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 $\mathrm{SU}(N)$ 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 $N$, two distinct types of SPT phases: an orbital Haldane phase, analogous to a spin-$N/2$ Haldane phase, and a topological $\mathrm{SU}(N)$ phase, which we fully characterize by its entanglement properties. We also propose sets of nonlocal order parameters that characterize the $\mathrm{SU}(N)$ topological phases found here.

DOI
10.1103/physrevb.91.075121

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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 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 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.
We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms. In particular, we focus on an SU(4) spin model [named SU(4) $e\text{\ensuremath{-}}g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling ng=3,ne=1{n}_{g}=3,{n}_{e}=1ng​=3,ne​=1 (ng,ne{n}_{g},{n}_{e}ng​,ne​: numbers of atoms in ground and excited states, respectively). Our numerical study with the infinite-size density matrix renormalization group shows that the ground state of SU(4) $e\text{\ensuremath{-}}g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by ${Z}_{4}\ifmmode\times\else\texttimes\fi{}{Z}_{4}$ symmetry. Specifically, we find that the ground state belongs to an SPT phase with the topological index $2\ensuremath{\in}{\mathbb{Z}}_{4}$ and show sixfold degenerate edge states. This is topologically distinct from SPT phases with the index $1\ensuremath{\in}{\mathbb{Z}}_{4}$ that are realized in the SU(4) bilinear model and the SU(4) Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We explore the phase diagram of SU(4) spin models including $e\text{\ensuremath{-}}g$ spin model, bilinear-biquadratic model, and AKLT model, and identify that antisymmetrization effect in neighboring spins (that we quantify with Casimir operators) is the driving force of the phase transitions. Furthermore, we demonstrate by using the matrix product state how the Z4{\mathbb{Z}}_{4}Z4​ SPT state with six edge states appears in the SU(4) $e\text{\ensuremath{-}}g$ spin model.
We investigate the nature of the Mott-insulating phases of half-filled 2N2N2N-component fermionic cold atoms loaded into a one-dimensional optical lattice. By means of conformal field theory techniques and large-scale DMRG calculations, we show that the phase diagram strongly depends on the parity of NNN. First, we single out charged, spin-singlet degrees of freedom that carry a pseudospin S=N/2\mathcal{S}=N/2S=N/2, making it possible to formulate a Haldane conjecture: For attractive interactions, we establish the emergence of Haldane insulating phases when NNN is even, whereas a metallic behavior is found when NNN is odd. We point out that the N=1,2N=1,2N=1,2 cases do not have the generic properties of each family. The metallic phase for NNN odd and larger than 1 has a quasi-long-range singlet pairing ordering with an interesting edge-state structure. Moreover, the properties of the Haldane insulating phases with even NNN further depend on the parity of N/2N/2N/2. In this respect, within the low-energy approach, we argue that the Haldane phases with N/2N/2N/2 even are not topologically protected but equivalent to a topologically trivial insulating phase and thus confirm the recent conjecture put forward by Pollmann et al. [arXiv:0909.4059 (to be published)].
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.
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.