Superconductivity and charge order of confined Fermi systems

The low-temperature properties of the two-dimensional attractive Hubbard model are strongly influenced by the fermion density. Away from half-filling, there is a finite-temperature transition to a phase with $s$-wave pairing order. However, $T_{\mathrm{c}}$ is suppressed to zero at half-filling, where long-range charge-density-wave order also appears, degenerate with superconductivity. This paper presents determinant quantum Monte Carlo simulations of the attractive Hubbard model in the presence of a confining potential $V_{\mathrm{trap}}$ which makes the fermion density $\rho$ inhomogeneous across the lattice. Pair correlations are shown to be large at low temperatures in regions of the trapped system with incommensurate filling, and to exhibit a minimum as the local density $\rho \left(\mathit{i}\right)$ passes through one fermion per site. In this ring of $\rho \left(\mathit{i}\right)=1$, charge order is enhanced. A comparison is made between treating $V_{\mathrm{trap}}$ within the local-density approximation (LDA) and in an ab initio manner. It is argued that certain sharp features of the LDA result at integer filling do not survive the proximity of doped sites. The critical temperature of confined systems of fixed characteristic density is estimated.

Figure 1

Fermion density $\rho$, kinetic energy $t$, and double occupancy $D$ as a function of distance $\left|\mathit{i}\right|$ from the trap center, with $N=564$ particles, $U=6$, $V_{\mathrm{trap}}=0.0097$, and $\mu =0.8$ at $\beta =9$. The solid line shows the LDA result; in this case, near perfect agreement is found between the true trapped system and the LDA. Note that, in contrast to the RHM, there is no plateau at half-filling (marked by the vertical line), which corresponds to the absence of a Mott gap in the homogeneous model. Error bars are smaller than the symbols.

Figure 2

Near-neighbor [$c_{\mathrm{pair}}(\mathit{i},\left(\begin{array}{c}1\\ 0\end{array}\right))$ ] and next-near neighbor [$c_{\mathrm{pair}}(\mathit{i},\left(\begin{array}{c}1\\ 1\end{array}\right))$ ] $s$-wave pair correlations as a function of distance $\left|\mathit{i}\right|$ from the trap center, with $N=564$ particles, $U=6$, $V_{\mathrm{trap}}=0.0097$, and $\mu =0.8$ at $\beta =9$. The solid lines show the LDA result; in this case, a striking failure of the LDA is seen around half-filling (marked by the vertical line). Due to CDW/$s$-wave degeneracy at half-filling in the homogeneous model, the LDA predicts superfluidity only sufficiently far from that point, while in the trapped system the superfluid phase may penetrate the half-filled ring. Error bars have been suppressed to avoid clutter, but the spread of the data points gives an indication of the uncertainties, which result from large statistical fluctuations observed at low $T$ (see text).

Figure 3

Near neighbor [$c_{\mathrm{pair}}(\mathit{i},\left(\begin{array}{c}1\\ 0\end{array}\right))$ ] and next-near neighbor [$c_{\mathrm{pair}}(\mathit{i},\left(\begin{array}{c}1\\ 1\end{array}\right))$, with the sign inverted for clarity] CDW correlations as a function of distance $\left|\mathit{i}\right|$ from the trap center, with $N=564$ particles, $U=6$, $V_{\mathrm{trap}}=0.0097$, and $\mu =0.8$ at $\beta =9$. The solid lines show the LDA result. As with the pair correlation, the LDA fails around half-filling (marked by the vertical line), where it predicts enhanced CDW correlations. Error bars are smaller than the symbols.

Figure 4

The $s$-wave structure factor $P_{\mathrm{S}}$ for systems of different linear size $L$ with $U=6$ as a function of inverse temperature $\beta$. At low temperatures ($\beta \gtrsim 7$), the curves flatten off to values determined by the system size, indicating a divergent correlation length.

Figure 5

The $s$-wave structure factor $P_{\mathrm{S}}$ as a function of linear lattice size $L$ for $U=6$ at various low temperatures (top to bottom: $\beta =\frac{1}{T}=16,15,...,10$) on a doubly logarithmic scale. A vertical shift has been applied to separate the curves from each other. A linear dependence on the doubly logarithmic scale (i.e., the structure factor varies as a power of the system size) is the expected behavior in the low temperature phase of a KT transition.

Figure 6

The effective critical exponent $\tilde{\eta }$ defined below Eq. (6), as obtained from nonlinear least-squares fits of the data from Fig. 4 to the functional form $P_{\mathrm{S}}(L;\tilde{\eta })\propto L^{2-\tilde{\eta }}$. The horizontal lines indicate the region $0⩽\tilde{\eta }⩽\frac{1}{4}$, expected for the low-temperature phase from the homogeneous case. Although the level of statistical noise is significant across all temperatures $T$, the exponent is consistent with values in this range for low temperatures.