Simulating a Mott Insulator Using Attractive Interaction

We study the particle-hole symmetry in the Hubbard model using ultracold fermionic atoms in an optical lattice. We demonstrate the mapping between charge and spin degrees of freedom and, in particular, show the occurrence of a state with “incompressible” magnetization for attractive interactions. Our results present a novel approach to quantum simulation by giving access to strongly correlated phases of matter through an experimental mapping to easier detectable observables.

Figure 1

Particle-hole transformation. (a) Mapping of site occupations. Under the transformation in Eq. (1), a singly occupied site with spin-up (-down) maps to a doubly occupied (empty) site and vice versa. (b) Mapping of many-body states. The paramagnetic Mott insulator transforms into a phase with local pairs, where attractive interactions favor the formation of pairs (doubles). Similarly, the spin ordering in the antiferromagnet transforms into a density ordering of the charge density wave. (c) Mapping of schematic phase diagrams. (left) The repulsive Hubbard model supports a wide range of strongly correlated phases, which can be particle-hole transformed to phases in the attractive Hubbard model (right). We note that the role of doping $\tilde{n}$ interchanges with the role of spin imbalance, i.e., magnetization $m$.

Figure 2

Particle-hole symmetry for various interaction strengths. Doubles (squares) and singles (circles) site occupation at $\mu =h=0$ vs $U/t$. For each interaction, we show the lowest temperature reached in our datasets. The red (blue) shaded regions show the results of the DQMC numerical calculations over $0.8\le k_{B}T/t\le 2$.

Figure 3

Interchanging the role of effective Zeeman field with the chemical potential. (a) For the attractive, spin-balanced ($h=0$) Hubbard model ($U/t=-7.6$) we show site occupations of singles (green circles) and doubles (black circles) at temperature $k_{B}T/t=1.5(1)$ as a function of $\mu$. (b) For the repulsive Hubbard model ($U/t=7.6$) with a finite spin imbalance ($h\ne 0$) we show site occupation of the majority or minority spin component (red or blue markers) as a function of both $h$ and $\mu$. Temperatures are $k_{B}T/t=1.4(1)$. The white plane indicates half filling $\mu =0$. The data points intersecting this plane are highlighted with squares and diamonds. (c) Combining datasets from (a) and (b) within a similar temperature range ($k_{B}T/t\approx 1.5$) leads to a collapse of the data onto the same curve.

Figure 4

Observing the Mott-like incompressibility with attractive interactions. Spin susceptibility ${\chi }_s$ (blue squares) at half filling and attractive interactions compared to the compressibility $\kappa$ (red circles) in a spin-balanced system with repulsive interactions [14]. We measure the magnetization $m$ at half filling for a large range of the effective Zeeman field $h$ up to $4.7t$. The spin susceptibility ${\chi }_s$ is obtained by performing the numerical derivative $\partial m/\partial h$. The error bars represent the standard errors. The range over which compressibility data $\kappa$ is presented is determined by the available atom number in our trap. In the case of the spin susceptibility ${\chi }_s$, we are limited by the temperature we can reach, while maintaining a considerable spin imbalance.