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arXiv:1307.7558 · Physical Review B

Two-orbital physics of high-spin fermionic alkaline-earth atoms confined in a one-dimensional chain

E. Szirmai

Published Nov 26, 2013• 7 citations
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Abstract

We study the effect of the coupling between the electronic ground state of high-spin alkaline-earth fermionic atoms and their metastable optically excited state, when the system is confined in a one-dimensional chain, and show that the system provides a possible realization of a finite momentum pairing (Fulde-Ferrell-Larkin-Ovchinnikov-like) state without spin- or bare mass imbalance. We determine the $\ensuremath{\beta}$-functions of the renormalization group trajectories for general spin and analyze the structure of the possible gapped and gapless states in the hydrodynamic limit. Due to the SU($N$) symmetry in the spin space, complete mode separation can not be observed even in the fully gapless 2$N$-component Luttinger liquid state. Contrary, four velocities characterize the system. We solve the renormalization group equations for spin-9/2 strontium-87 isotope and analyze in detail its phase diagram. The fully gapless Luttinger liquid state does not stabilize in the two-orbital system of the ${}^{87}$Sr atoms, instead, different gapped non-Gaussian fixed points are identified either with dominant density or superconducting fluctuations. The superconducting states are stable in a nontrivial shaped region in the parameter space as a consequence of the coupling between the two electronic states.

DOI
10.1103/physrevb.88.195432

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We study the effect of the coupling between the electronic ground state of high-spin alkaline-earth fermionic atoms and their metastable optically excited state, when the system is confined in a one-dimensional chain, and show that the system provides a possible realization of a finite momentum pairing (Fulde-Ferrell-Larkin-Ovchinnikov-like) state without spin- or bare mass imbalance. We determine the \ensuremathβ\ensuremath{\beta}\ensuremathβ-functions of the renormalization group trajectories for general spin and analyze the structure of the possible gapped and gapless states in the hydrodynamic limit. Due to the SU(NNN) symmetry in the spin space, complete mode separation can not be observed even in the fully gapless 2NNN-component Luttinger liquid state. Contrary, four velocities characterize the system. We solve the renormalization group equations for spin-9/2 strontium-87 isotope and analyze in detail its phase diagram. The fully gapless Luttinger liquid state does not stabilize in the two-orbital system of the 87{}^{87}87Sr atoms, instead, different gapped non-Gaussian fixed points are identified either with dominant density or superconducting fluctuations. The superconducting states are stable in a nontrivial shaped region in the parameter space as a consequence of the coupling between the two electronic states.
Recent experiments with Yb-173 and Sr-87 isotopes provide new possibilities to study high spin two-orbital systems. Within these experiments part of the atoms are excited to a higher energy metastable electronic state mimicking an additional internal (orbital) degree of freedom. The interaction between the atoms depends on the orbital states, therefore four different scattering channels can be identified in the system characterized by four independent couplings. When the system is confined into a one-dimensional chain the scattering lengths can be tuned by changing the transverse confinement, and driven through four resonances. Using the new available experimental data of the scattering lengths we analyze the phase diagram of the one-dimensional system as the couplings are tuned via transverse confinement, and the populations of the two orbital states are changed. We found that three orders compete showing power law decay: a state with dominant density wave fluctuations, another one with spin density fluctuations, and a third one characterized by exotic Fulde-Ferrell-Larkin-Ovchinnikov-like pairs consisting one atom in the electronic ground state and one in the excited state. We also show that sufficiently close to the resonances the compressibility of the system starts to diverge indicating that the emerging order is unstable and collapses to a phase separated state with a first order phase transition.
This dissertation focuses on a theoretical study of interacting electrons in one dimension. The research elucidates the ground state (zero temperature) electronic phase diagram of an aluminum arsenide quantum wire which is an example of an interacting one dimensional electron liquid. Using one dimensional field theoretic methods involving abelian bosonization and the renormalization group we show the existence of a spin gapped quantum wire with electronic ground states such as charge density wave and singlet superconductivity. The superconducting state arises due to the unique umklapp interaction present in the aluminum arsenide quantum wire bandstructure discussed in this dissertation. It is characterized by Cooper pairs carrying a finite pairing momentum. This is a realization of the Fulde-Ferrell-Larkin- Ovchinnikov state which is known to lead to inhomogeneous superconductivity. The dissertation also presents a theoretical analysis of the finite temperature single hole spectral function of the one dimensional electron liquid with gapless spin and charge modes (Luttinger liquid). The hole spectral function is measured in angle resolved photoemission spectroscopy experiments. The results predict a kink in the effective electronic dispersion of the Luttinger liquid. A systematic study of the temperature and interaction dependence of the kink provides an alternative way to detect spincharge separation in one dimensional systems where the peak due to the spin part of the spectral function is suppressed.
A two-leg ladder with nnn-component fermionic fields in the chains has been considered using an analytic renormalization group method. The fixed points and possible phases have been determined for generic filling as well as for a half-filled system and for the case when one of the subbands is half filled. A weak-coupling Luttinger-liquid phase and several strong-coupling gapped phases have been found. In the Luttinger liquid phase, for the most general spin dependence of the couplings, all 2n2n2n modes have different velocities if the interband scattering processes are scaled out, while nnn doubly degenerate modes appear if the interband scattering processes remain finite. The role of backward scattering, charge transfer, and umklapp processes has been analyzed using their bosonic form and the possible phases are characterized by the number of gapless modes. As a special case the SU(n)\mathrm{SU}(n)SU(n) symmetric Hubbard ladder has been investigated numerically. It was found that this model does not scale to the Luttinger liquid fixed point. Even for generic filling gaps open up in the spectrum of the spin or charge modes, and the system is always insulator in the presence of umklapp processes.
The formation of bound states of fermions in one dimension has always been one of the key topics in condensed matter physics. Motivated by recent experimental progresses in Prof. Jeremy Levy's group, we study the interplay of both species (spin and transverse band index) and mass imbalance in a mixture of two or more species of fermions with attractive interactions in one dimension. Previous theoretical and experimental efforts have shown the existence of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase for the case of two species with equal mass, in addition to the fully paired and fully polarized phases. For the unequal mass case, there are signatures of trion phases as well. We use DMRG to explore the rich possibilities of quantum phases and their transport signatures for the cases of two and more species of Fermions as we vary the interaction strengths and mass imbalances. With this we can gain insights into ongoing experiments with sketched nanowires in LAO/STO and ultracold atoms confined to one-dimensional tubes. We also study the formation of bound states in a single component Fermi chain with attractive interactions. The phase diagram, computed from DMRG (density matrix renormalization group), shows not only a superfluid of paired fermions (pair phase) and a liquid of fermion triplets (trion phase), but also a phase with two gapless modes. We show that the latter phase is described by a 2-component Tomonaga-Luttinger liquid (TLL) theory, consisting of one charged and one emergent neutral mode. We argue based on our numerical data, that the single, pair, and trion phases are descendants of the 2-component TLL theory. We speculate on the nature of the phase transitions amongst these phases.
We study the properties of a one-dimensional (1D) gas of fermions trapped in a lattice by means of the density matrix renormalization group method, focusing on the case of unequal spin populations, and strong attractive interaction. In the low-density regime, the system phase separates into a well-defined superconducting core and a fully polarized metallic cloud surrounding it. We argue that the superconducting phase corresponds to a 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, with a quasicondensate of tightly bound bosonic pairs with a finite center-of-mass momentum that scales linearly with the magnetization. In the large density limit, the system allows for four phases: in the core, we either find a Fock state of localized pairs or a metallic shell with free spin-down fermions moving in a fully filled background of spin-up fermions. As the magnetization increases, the Fock state disappears to give room for a metallic phase, with a partially polarized superconducting FFLO shell and a fully polarized metallic cloud surrounding the core.
We investigate properties of exotic pairing states in a three-dimensional Fermi gas with three-dimensional spin-orbit coupling and an effective Zeeman field. The interplay of spin-orbit coupling, effective Zeeman field, and pairing can lead to first-order phase transitions between different phases, and to interesting nodal superfluid states with gapless surfaces in the momentum space. We then demonstrate that pairing states with zero center-of-mass momentum are unstable against Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, with a finite center-of-mass momentum opposite to the direction of the effective Zeeman field. Unlike conventional FFLO states, these FFLO states are induced by the coexistence of spin-orbit coupling and Fermi surface deformation, and have intriguing features like first-order transitions between different FFLO states, nodal FFLO states with gapless surfaces in momentum space, and exotic fully gapped FFLO states. With the recent theoretical proposals for realizing three-dimensional spin-orbit coupling in ultracold atomic gases, our work is helpful for future experimental studies and provides valuable information for the general understanding of pairing physics in spin-orbit-coupled fermionic systems.