Cooling a band insulator with a metal: fermionic superfluid in a dimerized holographic lattice

Abstract

A cold atomic realization of a quantum correlated state of many fermions on a lattice, eg. superfluid, has eluded experimental realization due to the entropy problem. Here we propose a route to realize such a state using holographic lattice and confining potentials. The potentials are designed to produces a band insulating state (low heat capacity) at the trap center, and a metallic state (high heat capacity) at the periphery. The metal "cools" the central band insulator by extracting out the excess entropy. The central band insulator can be turned into a superfluid by tuning an attractive interaction between the fermions. Crucially, the holographic lattice allows the emergent superfluid to have a high transition temperature - even twice that of the effective trap temperature. The scheme provides a promising route to a laboratory realization of a fermionic lattice superfluid, even while being adaptable to simulate other many body states.

Figure 1

Concept: (a) Stage 1: Fermion gas(with equal number of ↑ and ↓ spins) in a confining potential(V_conf) (without the lattice pattern); here fermion dispersion is qualitatively free particle like (purple curve).(b) Stage 2: The lattice pattern is adiabatically ramped up and superimposed on the existing confining potential, which results in a dispersion with conduction(blue) and valence(black) bands. The confining potential profile is designed to tune the local chemical potential such that the valence band is completely occupied near the trap center(band insulating state) and only partially occupied(metallic state) near the trap peripheries. This results in a flow of entropy(green arrow) from the center to the periphery; the metal “cools” the band insulator. The system around the trap center, in the band insulating state with one particle per site, has exponentially low entropy. (c) Stage 3: Adiabatically turning on attractive interaction causes the band insulator to form a high transition temperature superfluid by pairing of opposite spin fermions.

Figure 2. Proposed Setup: Hologram H1 modulates Beam 1, to encode the lattice pattern and hologram H2 encodes the confining potential into Beam 2.

Beam splitter(BS) combines the outputs from H1 and H2. Lenses L1–3 stand for the optics appropriate for obtaining the required potential at F.

Figure 3. Dimerized Holographic Lattice:

(a) Lattice pattern of the dimerized holographic lattice with the tight binding model overlayed on top. Double lines are dimer bonds with hopping t_d. Single line slanted bonds have t and single line horizontal bonds have t_p hoppings. , are lattice basis vectors. The distances a, b, c can be chosen by a suitable design of the hologram to obtain desired values of t/t_d and t_p/t_d. (b) Conduction(C) and valence(V) band dispersions showing the energy gap; blue(C) and black(V) bands correspond to that shown in fig. 1(b). The gap occurs at ±k₀ along k₁ = k₂, as shown by the dashed line in the Brillouin zone plot(inset) of ε_C − ε_V (see equation (3)). Holographic lattice parameter values are V/E_R = 5.0, w/a = 0.3 and b/a = 1.04. The area around ±k₀ is highlighted with lighter contours. The resulting tight binding model has t/t_d = t_p/t_d = 0.64. (c) One particle density of states around the band gap, showing the van Hove singularities. (d) Dependence of wavevector k₀ (at which peaks in time-of-flight images are expected) on t/t_d. , are reciprocal lattice vectors.

Figure 4. Confining Potential:

(a) Radial profile of the confining potential (see equation (4)). The lengths l_h and r_d can be tuned to adjust the relative size of band insulator and metallic regions. (b) Entropy density s(r) and number density per site n(r) (inset) plots, for s₀ = 0.4. The lattice potential considered corresponds to that discussed in fig. 3. The trap temperature T/t_d ≈ 0.057. Confining potential parameters are V_h/E_R = 0.0045, l_h/a = 450, r_d/a = 1700.

Figure 5. Temperature Regimes of the Band Insulator Superfluid:

Plot of T_P (dashed red line), the temperature below which pairing of fermions is favored and T_BKT (solid blue line), the Beresinskii-Kosterlitz-Thouless temperature below which superfluidity emerges, as a function of t/t_d, for U/t_d = 3.0.