Finite-size scaling at first-order quantum transitions when boundary conditions favor one of the two phases

We investigate scaling phenomena at first-order quantum transitions, when the boundary conditions favor one of the two phases. We show that the corresponding finite-size scaling behavior, arising from the interplay between the driving parameter and the finite size of the system, is more complex than that emerging when boundary conditions do not favor any phase. We discuss this issue in the framework of the paradigmatic one-dimensional quantum Ising model, along its first-order quantum transition line driven by an external longitudinal field. Specifically, three regions with distinct scaling behaviors emerge, which correspond to different values of the field (small, intermediate, and large field), according to its capability to modify the phase favored by the boundary conditions.

Figure 1

Magnetization profile along the magnetic-field direction in the presence of EFBC, for $L=40$, $g=0.8$, and two values of the longitudinal field $h$. Note that the local magnetization for $x=0$ and $x=L+1=41$ is exactly $-1$, because of the boundary conditions.

Figure 2

Central magnetization $M_c$ (a) and energy gap $\mathrm{\Delta }$ (b) in the Ising model with EFBC, as a function of the longitudinal field $h$, for fixed transverse field $g=0.8$. The various data sets correspond to different system sizes, as indicated in the legend. The continuous vertical lines and arrows denote the magnetic fields $h_{\mathrm{tr}}\left(L\right)$, which correspond to the minimum of the energy gap, as displayed in panel (b).

Figure 3

Energy gap ratio $\mathrm{\Delta }(L,h)/{\mathrm{\Delta }}_0\left(L\right)$ as a function of the rescaled variable $hL$, for $g=0.5$ (main frame) and for $g=0.8$ (inset). The symbols correspond to different values of $L$.

Figure 4

Minimum gap ${\mathrm{\Delta }}_m$ for $h=h_{\mathrm{tr}}\left(L\right)$ as a function of the system size $L$, for two different values of $g$, as explained in the legend. The dotted lines are only meant to guide the eye: They show that the data approximately behave as ${\mathrm{\Delta }}_m\sim e^{-bL}$.

Figure 5

Longitudinal-field value $h_{\mathrm{tr}}\left(L\right)$ at which the system exhibits a minimum in the gap, as a function of the size $L$, for two different values of $g$, as explained in the legend. The dashed lines correspond to the asymptotic behavior $h_{\mathrm{tr}}\left(L\right)\approx \eta /L$, cf. Eq. (21). The inset displays $L{h}_{\mathrm{tr}}\left(L\right)$ for $g=0.5$, as a function of $L^{-2/3}$: Data fall on a straight line, showing the presence of corrections of order $L^{-2/3}$. The uncertainty on the estimates of $h_{\mathrm{tr}}\left(L\right)$ is always smaller than the symbols sizes.

Figure 6

Ratio $\mathrm{\Delta }/{\mathrm{\Delta }}_m$, where $\mathrm{\Delta }\equiv \mathrm{\Delta }(L,h)$ is the gap and ${\mathrm{\Delta }}_m\equiv \mathrm{\Delta }(L,h_{\mathrm{tr}})$, as a function of the scaling variable $y$ defined in Eq. (23), for several system sizes (see the legend). Panel (a) is for $g=0.5$, while panel (b) is for $g=0.8$. We also report (continuous black curve) the two-level prediction (27).

Figure 7

Local magnetization at the center of the chain $M_c$, as a function of the scaling variable $y$, for several system sizes (see the legend). Panel (a) is for $g=0.5$, while panel (b) is for $g=0.8$. We also report (continuous black curve) the two-level prediction (28), with the same constant $c$ as in Fig. 6.

Figure 8

Average magnetization $M$, as a function of the scaling variable $y$, for several system sizes (see the legend) and $g=0.5$. We also report (continuous black curve) the two-level prediction (29), where the constant $c$ is the same as in Fig. 6. In the inset we report $M$ as a function of $h$.

Figure 9

The local central magnetization $M_c$ in the large-$h$ region, for fixed $g=0.8$ and different system sizes, as indicated in the legend.

Figure 10

Magnetization profile close to the boundaries, for $g=0.8$ and several values of $h$, as indicated in the legend. Data are all for $L=180$ (filled symbols), except empty red squares, which stand for $L=100$ and $h=0.008$ and are superposed to filled red squares (on the scale of the figure). The inset shows data collapse, after a rescaling of the $x$ position by a factor $\sqrt{h}$.

Figure 11

Relative length $v_{-}=(1-M)/4$ of the negatively magnetized phase, as a function of the rescaled variable $1/\left(h^{1/2}L\right)$. The various data sets are for different system sizes $L$, while the two panels are for $g=0.5$ (a) and for $g=0.8$ (b). Dashed straight lines are constrained fits of the numerical data at large $L$ to Eq. (32).

Figure 12

Energy gap $\mathrm{\Delta }(L,h)$ versus $h$ in the large-$h$ region, for $g=0.8$ and different system sizes, as indicated in the legend. The inset displays the same data versus $h^{2/3}$. The dependence $\mathrm{\Delta }\sim h^{2/3}$, see Eq. (34), emerges quite clearly.

Figure 13

The energy difference (gap) of the lowest states as a function of the localized magnetic field $h_l$ at the center of the chain, for $g=0.5$. The insets show its minimum ${\mathrm{\Delta }}_m\left(L\right)$ (top inset) and the corresponding $h_{l,\mathrm{tr}}\left(L\right)$ (bottom inset), which appears to approach a constant value with $O\left(L^{-1}\right)$ corrections. Indeed, by fitting the available data to $h_{l,\mathrm{tr}}\left(L\right)=h_{l,\mathrm{tr}}^{*}+b/L$, we obtain $h_{l,\mathrm{tr}}^{*}=1.02$, see the dashed line in the corresponding inset.

Figure 14

Renormalized average magnetization $M(L,h_l)$ for $g=0.5$ versus $y_l$ [see Eq. (50)] for the model with a local magnetic field at the center of the chain. The curves appear to approach a scaling function ${\mathcal{M}}_l\left(y_l\right)$ with increasing $L$, i.e., $M(L\to \infty ,h_l)\approx {\mathcal{M}}_l\left(y_l\right)$. Note that, although the magnetization appears positive for $y_l>0$, its large-$L$ limit is consistent with negative extrapolations, and ${\mathcal{M}}_l(y_l\to \infty )=0$.