Trap effects and the continuum limit of the Hubbard model in the presence of a harmonic potential

We discuss how to perform the continuum limit of the $d$-dimensional Hubbard model in the presence of a harmonic confining potential at zero temperature and fixed particle number. While for $d\ge 3$ the system can be mapped into a noninteracting two-component Fermi gas in a harmonic trap, for $d=1$ and 2 the two-body interaction is described by a Dirac $\delta$. We show that the properties of this continuum limit can be put in one-to-one correspondence with those obtained by applying the trap-size scaling formalism to the confined Hubbard model in the so called dilute regime (fixed number of particles and weak confinement). The correspondence for $d<3$ has been tested, in the case of two particles with opposite spin, by comparing numerical exact-diagonalization results of the lattice system with those obtained in the continuum limit.

Figure 1

Repulsive case: interpolation of $\sqrt{l}\rho (x=0,U_r=100)$ (b) and $\sqrt{l}\rho (x=0,U_r=10)$ (a) using the function $f\left(l\right)=a+b{l}^{-1}$ (solid blue line) to estimate the value of the parameter $a=\mathcal{R}(X=0,U_r)$ (dashed black line). The values of $\mathcal{R}(X=0,U_r=100)$ and $\mathcal{R}(X=0,U_r=10)$ have been reported in Table 1.

Figure 2

Attractive case: interpolation of $\sqrt{l}\rho (x=0,U_r=-10)$ (a) and $\sqrt{l}\rho (x=0,U_r=-100)$ (b) using the function $f\left(l\right)=a+b{l}^{-1}$ (solid blue line) to estimate the value of the parameter $a=\mathcal{R}(X=0,U_r=10)$ (dashed black line). The values of $\mathcal{R}(X=0,U_r=-10)$ and $\mathcal{R}(X=0,U_r=-100)$ have been reported in Table 1.

Figure 3

TSS behavior of $\sqrt{l}\rho (x,U_r=100)$ for different values of the trap size $l$ and behavior of the one-body density function ${\rho }_{\text{GY}}(X,\alpha \left(U_r\right))$ (red dots).

Figure 4

TSS behavior of $\sqrt{l}\rho (x,U_r=10)$ for different values of the trap size $l$ and behavior of the one-body density function ${\rho }_{\text{GY}}(X,\alpha \left(U_r\right))$ (red dots).

Figure 5

TSS behavior of $\sqrt{l}\rho (x,U_r=-100)$ for different values of the trap size $l$ and behavior of the one-body density function ${\rho }_{\text{GY}}(X,\alpha \left(U_r\right))$ (red dots).

Figure 6

TSS behavior of $\sqrt{l}\rho (x,U_r=-10)$ for different values of the trap size $l$ and behavior of the one-body density function ${\rho }_{\text{GY}}(X,\alpha \left(U_r\right))$ (red dots).

Figure 7

TSS behavior of the double occupancy $l{d}_0(x,U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$), compared with the scaling function $d_{0\text{GY}}(X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=-10$ (red dots).

Figure 8

TSS behavior of the double occupancy $l{d}_0(x,U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$), compared with the scaling function $d_{0\text{GY}}(X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=10$ (red dots).

Figure 9

TSS behavior of the pair correlation $lP(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $P_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=-10$ (red dots).

Figure 10

TSS behavior of the pair correlation $lP(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $P_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=10$ (red dots).

Figure 11

TSS behavior of the one-particle correlation $\sqrt{l}C(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $C_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=-10$ (red dots).

Figure 12

TSS behavior of the one-particle correlation $\sqrt{l}C(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $C_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=10$ (red dots).

Figure 13

TSS behavior of the density-density correlation $lG(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $G_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=-10$ (red dots).

Figure 14

TSS behavior of the density-density correlation $lG(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $G_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=10$ (red dots).

Figure 15

TSS behavior of the spin-up–spin-down density correlation $lM(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $M_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=-10$ (red dots).

Figure 16

TSS behavior of the spin-up–spin-down density correlation $lM(0,x;U_r)$ for different values of the trap size $l$ ($l=1000,2000,5000$) with a point fixed at the trap center, compared with the scaling function $M_{\text{GY}}(0,X;\alpha \left(U_r\right))$ of the two-body unpolarized problem for $U_r=10$ (red dots).

Figure 17

Numerical results obtained for the rescaled density function $l\rho (\mathbf{x},U)$ compared with the one-body density function of the two-dimensional noninteracting problem ($U=0$, red solid line).

Figure 18

Numerical results for the rescaled density $l\rho (\mathbf{x},U=1)$ ($l=10,20,30$) compared with the one-body density associated with the ground-state wave function in correspondence of ${\alpha }_{2d}=1/2$.

Figure 19

Numerical results for the rescaled density $l\rho (\mathbf{x},U=3)$ ($l=10,20,30$) compared with the one-body density associated with the ground-state wave function in correspondence of ${\alpha }_{2d}=3/2$.