Numerical studies of ground-state fidelity of the Bose-Hubbard model

We compute ground-state fidelity of the one-dimensional Bose-Hubbard model at unit filling factor. To this aim, we apply the density matrix renormalization group algorithm to systems with open and periodic boundary conditions. We find that fidelity differs significantly in the two cases and study its scaling properties in the quantum critical regime.

Figure 1

Fidelity in an open Bose-Hubbard chain for different system sizes and the parameter shift $\delta =0.02$.

Figure 2

Fidelity in a periodic Bose-Hubbard model for different system sizes and $\delta =0.02$.

Figure 3

Black solid line shows fidelity in a periodic chain, while the dashed red line shows fidelity in an open chain. In both cases, $M=64$ and $\delta =0.02$.

Figure 4

The minimum of fidelity as a function of the system size (open boundary conditions). Pluses come from numerics done for $\delta =0.02$ and $M=32,64,128,256,512,1024,2048$. The line represents Eq. (6) with $\alpha \simeq 2.51$ obtained from the fit.

Figure 5

Scaling of the minimum of fidelity with the system size $M$ and the parameter shift $\delta$ (open boundary conditions). (a) Pluses come from numerics done for $M=32,64,128,256,512,1024,2048$ [38]. The line represents the fit: $ln\left[-lnF\left(J^{*}/U\right)\right]=-7.142+1.032lnM$. (b) Pluses come from numerics done for $\delta =0.005,0.01,0.02,0.03,0.04$ [38], while the line represents the fit: $ln\left[-lnF\left(J^{*}/U\right)\right]=7.833+1.994ln\delta$. Both fits accurately support the key features of Eq. (6): $lnF\left(J^{*}/U\right)\sim -M$ tested in panel (a) and $lnF\left(J^{*}/U\right)\sim -{\delta }^2$ tested in panel (b).

Figure 6

Position of the minimum of fidelity as a function of the system size (open boundary conditions). $\times$'s show numerics done for $M=32,64,128,256,512,1024,2048$ and $\delta =0.02$ [38]. The red (dashed) line is the fit (7) while the blue (solid) corresponds to (8). The inset shows the zoom for large $M$ to enable comparison of both fits.

Figure 7

Fidelity per site for $\delta =0.02$ and three system sizes: $M=256,512,1024$ (black solid, red dashed, green dotted-dashed lines, respectively). This plot is made from data from the lower panel of Fig. 1.

Figure 8

Fidelity susceptibility per site (open boundary conditions). The black solid, red dashed, green dotted-dashed lines represent data for $M=256,512,1024$, respectively. The inset enlarges the part of the main plot, where the curves cross.

Figure 9

Scaling of the minimum of fidelity with the system size $M$ and the parameter shift $\delta$ in a periodic Bose-Hubbard chain. Upper panel: $\times$'s show numerics for $M=16,32,48,64$ [38]. The line represents the fit $ln\left[-lnF\left(J^{*}/U\right)\right]=-6.787+0.998lnM$. Lower panel: $\times$'s show numerics for $\delta =0.005,0.01,0.02,0.03,0.04$. The line represents the fit $ln\left[-lnF\left(J^{*}/U\right)\right]=5.104+1.982ln\delta$.

Figure 10

Position of the minimum of fidelity as a function of the system size $M$ in the periodic Bose-Hubbard model. Stars show numerics done for $\delta =0.02$ [38], while the line represents Eq. (7) with $J^{*}\left(\infty \right)/U\simeq 0.27$, $a\simeq 0.33$, and $b\simeq 0.44$ (all coming from the fit).

Figure 11

Fidelity susceptibility per site (periodic boundary conditions). The black solid, red dashed, green dotted-dashed lines represent data for $M=16,32,64$, respectively. Fidelity susceptibility is obtained through the extrapolation procedure discussed around Eq. (9).

Figure 12

Density of atoms $\langle {\widehat{n}}_i\rangle$ in the ground state of the open-chain Bose-Hubbard model computed for $M=256$ and $J/U=0.21$, i.e., near the minimum of fidelity (Fig. 1).