XXZ spin-$\frac{1}{2}$ representation of a finite-$U$ Bose-Hubbard chain at half-integer filling

Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-$\frac{1}{2}$ representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy $U$. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-$\frac{1}{2}$ XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of $U$ and for a number $L$ of lattice sites as low as $L\sim 30$. We find an excellent agreement up to $10\%$ between the output of analytical and numerical computations, even for relatively small values of $U$. Our analysis implies that, also at finite $U$, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.

Figure 1

${\Delta }_{\mathrm{eff}}$ vs $J/U$ for different values of $V/J$ and $\overline{n}$. The top (bottom) dotted line corresponds to the value of ${\Delta }_{\mathrm{eff}}$ for $U/J\to \infty$ and $V/J=0.5$ ($V/J=0$). The other lines are for $V/J=0.5$ (top) and $V/J=0$ (bottom), with $\overline{n}=0$ (solid black lines), $\overline{n}=10$ (dashed red lines), and $\overline{n}\to \infty$ (dotted-dashed blue lines). Inset: same as in the main panel, but for $\eta$ vs $J/U$.

Figure 2

Density-density correlations $|\langle (n_i-f)(n_j-f)\rangle |$ vs $r=\left|i-j\right|$ for $U=10t$, $V=0.5t$, and $f=0.5$, with number of sites $L=150$. Black squares: numerical BH results; green diamonds: XXZ result with $a$, $b$, $c$ analytically determined; blue stars: $U\to \infty$ XXZ result [indicated by the label “infinite-$U$ limit” (see text fur further details)]. Lines are guide for eye. On the scale of the figure, results obtained for the XXZ model with $a$, $b$, $c$ numerically determined (not shown here) are indistinguishable from those obtained with the corresponding analytical values. Notice also the excellent agreement between numerical BH findings and analytical XXZ results.

Figure 3

Real part of $\langle b_i^{†}{b}_j\rangle$ vs $r=\left|i-j\right|$ for $U=10t$, $V=0.5t$, $f=0.5$, $L=150$. Magenta circles denote XXZ results with “nonrotated operators.” The notation for the other symbols is the same as in Fig. 2.

Figure 4

Relative error of the correlation function $|\langle (n_i-f)(n_j-f)\rangle |$ vs $r=\left|i-j\right|$ for the same parameters (and the same conventions for symbols and lines) of Fig. 2. We also plot the results obtained from the XXZ model with $a$, $b$, $c$ numerically determined as red triangles: numerical and analytical estimates for finite $U$ practically coincide. The average value, with $r_{\max }=3L/5$, is $0.06\pm 0.04$ for the finite-$U$ XXZ model and $2.9\pm 1.2$ for the infinite-$U$ XXZ model.

Figure 5

Relative error of the real part of $\langle b_i^{†}{b}_j\rangle$ vs $r=\left|i-j\right|$ for the same parameters (and the same conventions for symbols and lines) of Fig. 3. The average values, with $r_{\max }=3L/5$, are $0.025\pm 0.004$ (green diamonds: analytical XXZ results), $0.38\pm 0.08$ (blue stars: XXZ result in the infinite-$U$ limit), $0.26\pm 0.03$ (magenta circles: finite-$U$ result with nonrotated operators).

Figure 6

In each panel, we plot $|\langle (n_i-f)(n_j-f)\rangle |$ vs $r=|i-j|$ for different sizes: $L=30,50,80,100$ ($U=10t$, $V=0.5t$, $f=0.5$). Black squares denote the BH results, red triangles the XXZ results with $a$, $b$, $c$ numerically determined, and blue stars the infinite-$U$ XXZ results (we do not report the XXZ results with $a$, $b$, $c$ analytically determined since they practically coincide with the red triangles).

Figure 7

Real part of $\langle {\widehat{b}}_i^{†}{\widehat{b}}_j\rangle$ vs $r=\left|i-j\right|$ for the sizes and the parameters of Fig. 6.

Figure 8

Density-density correlations $|\langle (n_i-f)(n_j-f)\rangle |$ vs $r=|i-j|$ for different values of $U/t=20,10,5,3.3$, corresponding, respectively, to $J/U=0.1,0.2,0.4,0.6$ (with $V=0.5t$, $f=0.5$, $L=150$).

Figure 9

Real part of $\langle {\widehat{b}}_i^{†}{\widehat{b}}_j\rangle$ vs $r=\left|i-j\right|$ for the $U$ values and the parameters of Fig. 8.

Figure 10

Plot of ${(-1)}^{i+1}\langle n_i-f\rangle$ vs $i$ numerically computed in the BH chain for $U=10t$, $f=0.5$, and $L=150$ for different values of $V$: from top to bottom $V/t=3.3$ (black circles), $V/t=3.1$ (red squares), $V/t=2.9$ (green diamonds), and $V/t=2$ (blue stars) corresponding, respectively, to ${\Delta }_{\mathrm{eff}}=1.09,1.00,0.91,0.52$.

Figure 11

Plot of $\langle n_i-f\rangle$ vs $i$ numerically computed in the BH chain for $U=10t$, $f=0.5$, and $L=150$ for different values of $V$: $V/t=-1.3,-1.5,-1.6,-1.7$, corresponding, respectively, to ${\Delta }_{\mathrm{eff}}=-0.91,-1.00,-1.04,-1.08$.

Figure 12

Plot of the modulus of $\langle n_i-f\rangle$ vs $i$ computed in the BH chain for the same parameters of Fig. 11: $V/t=-1.3,-1.5,-1.6,-1.7$. The corresponding averages are $\approx 0.02,0.04,0.06,0.21$.