Mesoscopic effects in quantum phases of ultracold quantum gases in optical lattices

We present a wide array of quantum measures on numerical solutions of one-dimensional Bose- and Fermi-Hubbard Hamiltonians for finite-size systems with open boundary conditions. Finite-size effects are highly relevant to ultracold quantum gases in optical lattices, where an external trap creates smaller effective regions in the form of the celebrated “wedding cake” structure and the local density approximation is often not applicable. Specifically, for the Bose-Hubbard Hamiltonian we calculate number, quantum depletion, local von Neumann entropy, generalized entanglement or Q measure, fidelity, and fidelity susceptibility; for the Fermi-Hubbard Hamiltonian we also calculate the pairing correlations, magnetization, charge-density correlations, and antiferromagnetic structure factor. Our numerical method is imaginary time propagation via time-evolving block decimation. As part of our study we provide a careful comparison of canonical versus grand canonical ensembles and Gutzwiller versus entangled simulations. The most striking effect of finite size occurs for bosons: we observe a strong blurring of the tips of the Mott lobes accompanied by higher depletion, and show how the location of the first Mott lobe tip approaches the thermodynamic value as a function of system size.

Figure 1

The first Mott lobe of the Bose-Hubbard Hamiltonian. Quantum depletion for a system of interacting bosons on 51 lattice sites at $T=0$ as a function of chemical potential $\mu$ and hopping parameter $t$, both normalized to interaction strength $U$, obtained with time-evolving block decimation (TEBD) in imaginary time propagation, as discussed below. The white circular points represent best-converged results of density matrix renormalization group methods for the boundary of the Mott lobe in the thermodynamic limit [8].

Figure 2

Quantum measures for the Bose-Hubbard Hamiltonian. Shown is the density (first row), average local entropy of entanglement (second row), generalized entanglement or Q measure (third row), quantum depletion (fourth row), fidelity (fifth row), and fidelity susceptibility (sixth row). The panels show increasing numbers of sites from left to right: 3, 5, 10, 20, and 40.

Figure 3

Crossover estimators. First derivative (left column) and second derivative (right column) with respect to $t/U$ of various quantum measures for unit filling in the canonical ensemble. The vertical line corresponds to the known value of $t_c$.

Figure 4

Estimates of the crossover. Extrema of the first derivative of the depletion (squares), the fidelity susceptibility (triangles), and the second derivative of the Q measure for unit filling in the canonical ensemble (circles). All derivatives are with respect to $t/U$. The horizontal line again corresponds to the limit $t_c$. The curves are a guide to the eye.

Figure 5

Open versus periodic boundary conditions. Fidelity susceptibility for fixed $N=L$ in a comparative study of open boundary conditions (OBC) and periodic boundary conditions (PBCs); the latter can only be obtained via exact diagonalization (ED) in our present code, while OBCs can be compared directly with TEBD. We note that (1) PBCs have a substantially different fidelity for smaller systems, with a peak at much smaller $t/U$, and (2) TEBD and exact diagonalization match up to the $\mathcal{O}(\delta t^2)$ error from the Suzuki-Trotter decomposition for small systems, validating our code.

Figure 6

Quantum measures for the Fermi-Hubbard Hamiltonian in particle-hole symmetric form. Density (first row), generalized entanglement or Q measure (second row), charge-density wave correlation (third row), pairing correlation (fourth row), antiferromagnetic structure factor (fifth row), and fidelity susceptibility (sixth row). The panels show increasing numbers of sites from left to right: 3, 5, 9, 10, and 20.

Figure 7

Comparison of canonical ensemble (CE) and grand canonical ensemble (GCE). Shown is the Q measure (generalized entanglement) for the Bose-Hubbard Hamiltonian and $L=20$ sites. The CE is restricted to curves of fixed number difference. To obtain the images in Fig. 2, we interpolate the CE by taking the nearest fixed line and thus “filling in” the phase diagram. The GCE shows that this method obtains the correct result, thus overcoming the finite differencing error inherent to the CE: $\mu \approx E(N+1)-E(N)$. Small qualitative differences remain between the two ensembles, as can be observed by comparing to the corresponding panel (second row, fourth column) in Fig. 2.

Figure 8

Quantum measures for the Bose-Hubbard Hamiltonian in the canonical ensemble. This figure shows the actual data underlying Fig. 2; for smaller systems, the canonical ensemble provides very poor resolution in chemical potential, as can be observed in the first few columns. As the system size increases, the resolution in chemical potential goes up and the Mott lobe tip appears.

Figure 9

Increasing corrections to mean-field Gutzwiller ansatz. The same quantities as calculated in Fig. 6 for the Fermi-Hubbard Hamiltonian, but for 10 sites and in increasing values of $\chi$ from left to right: 1, 2, 4, 8. $\chi =1$ corresponds to the Gutzwiller mean-field approximation. The speckle structure observed in the vacuum region and for entanglement in the $\chi =1$ case is simply numerical noise associated with the calculation of small numbers.

Figure 10

Quantitative error analysis for increasing entanglement ($\chi$) in TEBD. Average error (left column) and maximum error (right column) for the Fermi-Hubbard Hamiltonian (top row) and Bose-Hubbard Hamiltonian (bottom row) as a function of increasing powers of $\chi$. For fermions, we show the quantum measures of density (squares), Q measure or generalized entanglement (triangles), and pairing correlations (circles); for bosons, we show the error in the Q measure (squares) and quantum depletion (triangles) in the grand canonical ensemble. Curves are a guide to the eye only.