Spin-dependent Hubbard model and a quantum phase transition in cold atoms

We describe an experimental protocol for introducing spin-dependent lattice structure in a cold atomic Fermi gas using lasers. It can be used to realize Hubbard models whose hopping parameters depend on spin and whose interaction strength can be controlled with an external magnetic field. We suggest that exotic superfluidities will arise in this framework. An especially interesting possibility is a class of states that support coexisting superfluid and normal components, even at zero temperature. The quantity of normal component varies with external parameters. We discuss some aspects of the quantum phase transition that arises at the point where it vanishes.

Figure 1

Spin-dependent lattice: two counterpropagating laser beams generate a standing light wave which leads to different ac Stark shifts for the spin-up and spin-down components of the ground state atoms $(\sigma =\uparrow ,\downarrow )$. As a result the tunneling matrix elements $t_{\sigma }$ for the two spin components are different. The spin-up and -down components are coupled by a Raman laser with Rabi frequency ${\Omega }_0$ and detuning $\delta$.

Figure 2

(a) Atomic level scheme of alkali-metal atoms. A ${\sigma }^{+}$ polarized lattice beam is tuned between the two excited state fine structure components resulting in different ac Stark shifts for the ground state levels as explained in the context of Eq. (4). (b) Two possible “spin-up” and “spin-down” states are illustrated the case of ${}^{40}\mathrm{K}$ atoms.

Figure 3

ac Stark shift (in arbitrary units) of the ${}^{40}\mathrm{K}$ hyperfine $M_F=9∕2$ (solid line) and $M_F=7∕2$ (dashed line) states [compare Fig. 2b] as a function of the detuning $\Delta$. The value of $\Delta =0$ corresponds to the ${}^{2}P_{1∕2}$ excited fine structure state, while $\Delta ={\Delta }_{\mathrm{fs}}$ is the ${}^{2}P_{3∕2}$ resonance. We note the strong spin dependence (i.e., dependence on the internal state) for detuning between the two fine structure states.

Figure 4

(a) ac Stark shift (in arbitrary units) of the $M_F=9∕2$ (solid line) and the $M_F=7∕2$ state (dashed line) in the detuning region right of the interference zero $0.248<\Delta ∕{\Delta }_{\mathrm{fs}}$ of the $M_F=7∕2$ state. (b) Ratio of the hopping matrix elements for the $t_{\uparrow \equiv \mid M_F=9∕2⟩,\ell }∕t_{\downarrow \equiv \mid M_F=7∕2⟩,\ell }$ for a given spatial direction $\ell =1,2,3$ as a function of detuning $\Delta$ for a fixed $V_{\uparrow \equiv \mid M_F=9∕2⟩,\ell }^{\left(0\right)}\equiv V_0$ with $V_0=5$ (curve $a$), 10 (curve $b$), and $20E_R$ (curve $c$) in units of the recoil energy $E_R={\hslash }^2{k}^2∕2m$. The hopping matrix elements were obtained from a band structure calculation for the given depth of the optical potential.

Figure 5

Quantum phase transition seen from change of quasiparticle spectra. The critical value $h=h_c$ is determined by requiring that the $E_2$ branch have just one solution at zero energy. For $h>h_c$, one should imagine a three-dimensional momentum space in which two surfaces, defined by ${\Theta }_{\mathbf{k}}={\Theta }^{-}$ and ${\Theta }_{\mathbf{k}}={\Theta }^{+}$, are gapless for the $E_2$ branch. (a) BCS superfluid; (b) quantum critical state; (c) “$\text{superfluid}+\text{normal}$” BP state. For convenience, $t_{\uparrow }⩾t_{\downarrow }$ is always assumed throughout the paper.

Figure 6

An illustration of changing topology of the spin-up Fermi sea. Shown are cross sections at $k_z=\pi ∕2a$ in a 3D first Brillouin zone for (a) BCS and (b) BP; momentum units for $k_x$, $k_y$ are $\pi ∕a$ where $a$ is the cubic lattice constant. As usual, the occupied states spread out over the entire Brillouin zone due to the pairing interaction. The shading in the graph indicates that the region has a nonvanishing occupation number. The dotted lines indicate the orginal free Fermi surfaces of spins up (inner curve) and spins down (outer curve). The solid lines in (b) are energy surfaces defined by ${\Theta }_{\mathbf{k}}={\Theta }^{-}$ and ${\Theta }_{\mathbf{k}}={\Theta }^{+}$; for small $\Delta \to 0$, they merge with the free Fermi surfaces (dotted lines). The phase transition occurs at the point in which a simply connected Fermi sea is isolated into two regions.

Figure 7

Quantum phase transition shown by spin polarization $M$. For the BCS (BP) state, $M=0$ $(M\ne 0)$.