Numerically exact mimicking of quantum gas microscopy for interacting lattice fermions

A numerical method is presented for reproducing fermionic quantum gas microscope experiments in equilibrium. By employing nested componentwise direct sampling of fermion pseudodensity matrices, as they arise naturally in determinantal quantum Monte Carlo (QMC) simulations, a stream of pseudosnapshots of occupation numbers on large systems can be produced. There is a sign problem even when the conventional determinantal QMC algorithm can be made sign-problem free, and every pseudosnapshot comes with a sign and a reweighting factor. Nonetheless, this “sampling sign problem” turns out to be weak and manageable in a large, relevant parameter regime. The method allows one to compute distribution functions of arbitrary quantities defined in occupation number space and, from a practical point of view, facilitates the computation of complicated conditional correlation functions. While the projective measurements in quantum gas microscope experiments achieve direct sampling of occupation number states from the density matrix, the presented numerical method requires a Markov chain as an intermediate step and thus achieves only indirect sampling, but the full distribution of pseudosnapshots after (signed) reweighting is identical to the distribution of snapshots from projective measurements.

Figure 1

Componentwise direct sampling in a single configuration of Hubbard-Stratonovich fields $\left\{\mathbf{s}\right\}$. (a) Lattice sites (red circles) are ordered boustrophedonically and the joint distribution of their occupation numbers is written as a chain of conditional probabilities. For each sampled component (i.e., lattice site $k$), the pseudoprobability distribution $p_{\left\{\mathbf{s}\right\}}\left(n_k\right)$ is reweighted such that samples can be drawn from a valid probability distribution $q\left(n_k\right)$. (b) Pseudosnapshot generated for a given Hubbard-Stratonovich field configuration. Doublon-hole (d-h) fluctuations, first revealed experimentally by bunching of “antimoments” [12], are clearly visible in this pseudosnapshot, which comes with a positive sign and a modest reweighting factor of $R\approx 1.22$. Square lattice with parameters $U/t=10,\beta t=10,\langle n\rangle =1,L=16$.

Figure 2

Joint distribution $P(M_{\text{stag}}^z,Q_{\text{stag}}^z)$ at half filling. (a) Weak Hubbard interaction $U/t=1,\beta t=4,L=12,L_A=12$. Even-odd oscillations, which are visible in the joint distribution, are smeared out in the marginal distributions (left and bottom of each panel). (b) Close to the metal-insulator crossover: $U/t=8,\beta t=5,L=12,L_A=12$. In the heat map outliers have been set to zero. (c) Strong Hubbard interaction: $U/t=14,\beta t=5,L=16,L_A=8$.

Figure 3

Three-point spin-charge correlations $C_{\left|\mathbf{d}\right|}\left(\right|\mathbf{r}\left|\right)$, as depicted in the insets, around an isolated hole (see main text). $U/t=14$, $\mu /t=-3$, $\beta t\in \{2,2.5\}$, system size $L\times L$ with $L=10$. A total number of ${10}^8$ pseudosnapshots has been generated with 224 independent Markov chains. For $\mu /t=-3$ and $L=10$, the average number of excess holes is $\langle N_h\rangle -\langle N_d\rangle \approx 0.35$ (see FCS of $N_h$ and $N_d$ in Appendix pp5). The data is consistent with data for comparable parameters of temperature and interactions ($\beta t=2.2$ and $\beta J=0.66$) from Ref. [29] where a single hole in the $t\text{−}J$ model was simulated.

Figure 4

Sampling sign problem at half filling. The probe area is the total $L\times L$ square system ($L=L_A$). (a) Average sign of a pseudosnapshot. (b) Average (unsigned) reweighting factor (full lines) and average maximum reweighting factor (dashed lines). The red dashed-dotted line indicates a rough estimate of the threshold of maximum reweighting factors below which numerically exact results can be obtained with modest computational effort.

Figure 5

Conventional sign problem and sampling sign problem away from half filling: dependence on $U/t$. (a),(c) Average sign ($\langle s\rangle$, full lines) and average sampling sign ($\langle \tilde{s}\rangle$, dashed lines); (b),(d) average reweighting factor (full lines) and average maximum reweighting factor (dotted lines). The inverse temperature $\beta$ is in units of $1/t$. The probe area is equal to the system size with $L_A=L=8$.

Figure 6

Conventional sign problem and sampling sign problem away from half filling: dependence on inverse temperature $\beta t$. System size $L_A=L=8$.

Figure 7

Probabilities $P\left(s\right)$ of all microstates $s$ in the occupation number basis. Comparison with exact diagonalization (ED) for the parameters shown in the upper panel and for the hopping matrix of Eq. (D2). The total number of Fock samples is ${10}^7$, generated by 56 independent Markov chains.

Figure 8

(a) Probabilities of spin-only states around an empty site as obtained from reweighted pseudosnapshots in the reference frame of the hole. Parameters: $U/t=12,\mu /t=-2,\beta t=2.5$, $L\times L$ square lattice with $L=8$. For this system size, the average number of holes and doubles $\langle N_h\rangle \approx 1.50$ and $\langle N_d\rangle \approx 1.31$ so that the average number of excess holes is $\langle N_h\rangle -\langle N_d\rangle \approx 0.19$. With 56 independent Markov chains a total number of $70\times {10}^6$ occupation number samples of the entire system was obtained. In the reference frame of the hole, those states around the hole position are filtered out which do not contain charge fluctuations (“spin-only states”). Per Hubbard-Stratonovich configuration, 200 direct sampling steps were performed during the Monte Carlo sweeps at different imaginary time slices. Trotter discretization $\mathrm{\Delta }\tau t=0.02$. (b) Illustration of the most important spin-only states around an isolated hole which are marked in panel (a). Dashed lines delineate different antiferromagnetic domains.

Figure 9

FCS of the total number of doubly occupied sites $\langle N_d\rangle$ and holes $\langle N_h\rangle$, obtained from reweighted pseudosnapshots for different levels of hole doping (top and bottom panel). Dashed and dotted lines are binomial distributions; the dashed-dotted line is a “shifted” binomial distribution. Parameters: $U/t=12$, $\beta t=4,L=8$.

Figure 10

Three-point spin-charge correlation function $C_{\left|\mathbf{d}\right|=2}\left(\right|\mathbf{r}\left|\right)$ of Eq. (F1) computed on a $4\times 4$ system (a)–(c) as well as a $6\times 6$ system (d)–(g) with periodic boundary conditions and with parameters $U/t=14,\beta t=2.5,\mu =-4.0,-4.5,...,-6.5$. The dashed magenta line in (c) [denoted “conditional $C_2$”] shows the correlation function $C_{\left|\mathbf{d}\right|=2}\left(\right|\mathbf{r}\left|\right)$ calculated for $\mu /t=-4.0$ according to Eq. (12) in the main text, in which only those snapshots are counted where the hole is surrounded on eight sites by spin-only states (i.e., it is not adjacent to a doublon or a hole and can be regarded as an isolated dopant). The experimental data of Ref. [19] are also shown for comparison. There, the system size is approximately $4\times 6$ to $6\times 6$ lattice sites (slightly inhomogeneous with open boundary conditions), interactions $U/t\approx 14$, and stated temperature $T=1.4J$, which corresponds to $T/t\approx 0.4$ (i.e., $\beta t=2.5$) for $J=4t^2/U$. In the quantum gas microscope experiment, there are on average 1.95(1) dopants present in each experimental realization (doublon doped instead of hole doped) [19]. For our setting on a $4\times 4$ system at $\mu /t=-4.0$, on the other hand, the average number of holes (hole doped) is $\langle N_h\rangle =0.72$, which is small enough for measuring the spin environment of an isolated dopant. (d) Randomly chosen pseudosnapshots for $\mu /t=-4$ (left) and $\mu /t=-5$ (right).

Figure 11

FCS of the number of doubly occupied ($N_d$) and empty sites ($N_h$) for the same parameters as in Fig. 3 of the main text, which are $L=10,U/t=14,\mu /t=-3.0,\beta t=2.5$.