Full Counting Statistics for Interacting Fermions with Determinantal Quantum Monte Carlo Simulations

We present a method for computing the full probability distribution function of quadratic observables such as particle number or magnetization for the Fermi-Hubbard model within the framework of determinantal quantum Monte Carlo calculations. Especially in cold atom experiments with single-site resolution, such a full counting statistics can be obtained from repeated projective measurements. We demonstrate that the full counting statistics can provide important information on the size of preformed pairs. Furthermore, we compute the full counting statistics of the staggered magnetization in the repulsive Hubbard model at half filling and find excellent agreement with recent experimental results. We show that current experiments are capable of probing the difference between the Hubbard model and the limiting Heisenberg model.

Figure 1

(a) Setup of the square subsystem $A$ of dimension $L_A\times L_A$ embedded in a larger $L\times L$ square lattice with periodic boundary conditions. (b)–(d) Particle number distribution $P_N(N_A)$ along the BCS-to-BEC crossover at half filling for $U/t=-1,-8,-14$. For comparison, the distribution for a BCS mean-field ansatz at $T=0$ is shown in red, as well as the Heisenberg model for strong interactions. (e) Broadening of the distribution function $P(N_A)$ and smearing of the even-odd oscillations for increasing subsystem size $L_A$.

Figure 2

(a) Scaling of the even-odd splitting $P_{\text{odd}}$ with linear subsystem size $L_A$. Blue dashed curves in (a) represent the fit function Eq. (12). (b),(c) Pair size $p_{\text{deloc}}$ as a function of attractive interaction $-|U|/t$ and filling $⟨n⟩$ in comparison with the predictions by mean-field theory in the BEC-BCS crossover. For clarity, data points in (c) for $⟨n⟩\ne 1$ are not reproduced again in (b). The inset in (b) shows the two-body bound state in vacuum at $U=-4$ (see main text); the corresponding $p_{\text{deloc}}$ is indicated by the solid red line in (b) and the red squares in (c). In the limit $⟨n⟩\to 0$, the pair size as computed with DQMC calculations approaches $p_{\text{deloc}}$ for the two-body bound state in vacuum at the same interaction $U$.

Figure 3

FCS of the staggered magnetization on a disk-shaped subsystem of $N_s=80$ sites for repulsive Hubbard interaction $U/t=7.2$ at half filling and for the same four temperatures (a)–(d) as in Ref. [18]. Red histograms are reproduced from Ref. [18]; blue error bars are our DQMC simulations. Binned FCS of the Heisenberg model at an equivalent temperature are shown in black. The magenta line in (d) is a simple model of independent sites [32].